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noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Riemann surfaces; inhomogeneous Cauchy-Riemann equation with \(L^{2}\) estimates; holomorphic line bundle with positive curvature; subharmonic exhaustion function; divisor; uniformization theorem; biholomorphic classification of Riemann surfaces; Teichmüller theory T.~Napier, M.~Ramachandran: {\em An Introduction to Riemann Surfaces}, Springer (2011). DOI 10.1007/978-0-8176-4693-6; zbl 1237.30001; MR3014916 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable, Compact Riemann surfaces and uniformization, Harmonic functions on Riemann surfaces, Differentials on Riemann surfaces, Conformal metrics (hyperbolic, Poincaré, distance functions), Teichmüller theory for Riemann surfaces, Riemann surfaces; Weierstrass points; gap sequences, Vector bundles on curves and their moduli, Relationships between algebraic curves and integrable systems An introduction to Riemann surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces value sets; finite fields; polynomials; towers of function fields Polynomials over finite fields, Algebraic functions and function fields in algebraic geometry A link between minimal value set polynomials and tamely ramified towers of function fields over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces ideals; Lie algebras of vector fields; derivation Lie algebras; affine varieties Siebert, T., Lie algebras of derivations and affine algebraic geometry over fields of characteristic 0, \textit{Math. Ann.}, 305, 271-286, (1996) Lie algebras of vector fields and related (super) algebras, Derivations, actions of Lie algebras, Lie (super)algebras associated with other structures (associative, Jordan, etc.), Infinite-dimensional Lie (super)algebras, Varieties and morphisms Lie algebras of derivations and affine algebraic geometry over fields of characteristic \(0\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces intersection multiplicity; Hilbert function; blow up; completed tensor product; integral closure of ideals; associated graded ring Skalit, C., Intersection multiplicity of Serre in the unramified case, J. Algebra, 428, 91-121, (2015) Homological conjectures (intersection theorems) in commutative ring theory, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Intersection multiplicity of Serre in the unramified case

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces singularities; analytic algebras; singularities of holomorphic vector fields and foliations Druel, S., The Zariski-lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc., 46, 827-835, (2014) Singularities in algebraic geometry, Analytic algebras and generalizations, preparation theorems, Singularities of holomorphic vector fields and foliations The Zariski-Lipman conjecture for log canonical spaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces involutions on Riemann surfaces; symmetric surfaces; automorphism groups; anticonformal automorphisms; conjugacy classes of symmetries; Klein surfaces Gromadzki, G., Izquierdo, M.: Real forms of a Riemann surface of even genus. Proc. Am. Math. Soc. 126(12), 3475--3479 (1998) Braid groups; Artin groups, Compact Riemann surfaces and uniformization, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Hyperbolic and elliptic geometries (general) and generalizations, Riemann surfaces; Weierstrass points; gap sequences, Klein surfaces, Topology of real algebraic varieties, Generators, relations, and presentations of groups Real forms of a Riemann surface of even genus

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symbolic power; sum of ideals; associated prime; tensor product; binomial expansion; depth; Castelnuovo-Mumford regularity; tor-vanishing; depth function Dimension theory, depth, related commutative rings (catenary, etc.), Singularities in algebraic geometry, Homological functors on modules of commutative rings (Tor, Ext, etc.), Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) Symbolic powers of sums of ideals

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic geometric codes; algebraic curves over finite fields; function fields; Goppa codes; complexity of multiplication in extension fields; divisors of curves of genus 1; weight distributions; minimal weight Other types of codes, Geometric methods (including applications of algebraic geometry) applied to coding theory, Finite ground fields in algebraic geometry, Algebraic functions and function fields in algebraic geometry Contributions to the theory of coding and complexity using algebraic function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces orders of the Brauer groups; diagonal surfaces; finite fields; K3 surfaces Finite ground fields in algebraic geometry, Brauer groups of schemes, \(K3\) surfaces and Enriques surfaces, Varieties over finite and local fields On the orders of the Brauer groups of some weighted diagonal surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces cohomology; dualities; schematic algebras; graded algebras; Serre's theorem; categories of quasicoherent sheaves; categories of graded modules; noncommutative projective schemes; graded rings; cohomological behaviour; Auslander-Gorenstein algebras (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Noncommutative algebraic geometry, Schemes and morphisms, Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings Schematic algebras and the Auslander-Gorenstein property

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Hasse-Weil bound; maximal curve; geometric Goppa code; asymptotically good sequence; survey; number of rational points; curves over finite fields; towers of function fields van der Geer, G., Curves over finite fields and codes, (European congress of mathematics, vol. II, Barcelona, 2000, Prog. math., vol. 202, (2001), Birkhäuser Basel), 225-238 Curves over finite and local fields, Algebraic coding theory; cryptography (number-theoretic aspects), Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry, Research exposition (monographs, survey articles) pertaining to number theory Curves over finite fields and codes

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces class of a projective algebraic surface; classification of algebraic surfaces; projective characters; ruled surfaces A. Lanteri, On the class of a projective algebraic surface. Arch. Math., 45 (1985), 79--85. Families, moduli, classification: algebraic theory, Special surfaces On the class of a projective algebraic surface

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces ruled degenerations of minimal ruled surfaces Rational and ruled surfaces, Formal methods and deformations in algebraic geometry, Families, moduli, classification: algebraic theory Correction to ``Degenerations of minimal ruled surfaces''

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces coverings of \(\mathbb{P}^ 1\); Galois groups; Hurwitz monodromy group; finite groups; Galois group of regular extension of \(\mathbb{Q}\); universal Frattini cover; rational point on varieties; regular realization problem on dihedral groups; real branch points; totally nonsplit extension; field of totally real numbers; fields of definition; inverse Galois problem; symmetric group; modular curves Dèbes, Pierre; Fried, Michael D., Nonrigid constructions in Galois theory, Pacific J. math., 163, 1, 81-122, (1994) Inverse Galois theory, Rational points, Varieties over global fields, Coverings of curves, fundamental group, Symmetric groups Nonrigid constructions in Galois theory

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symmetric tensor; sums of powers of linear polynomials; Waring problem; Waring rank; open Waring rank Jelisiejew, J.; An upper bound for the Waring rank of a form; Arch. Math.: 2014; Volume 102 ,329-336. Projective techniques in algebraic geometry, Multilinear algebra, tensor calculus, Canonical forms, reductions, classification, Polynomials, factorization in commutative rings, Vector spaces, linear dependence, rank, lineability, Determinants, permanents, traces, other special matrix functions An upper bound for the Waring rank of a form

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Picard modular surfaces; Shimura varieties; Fuchsian systems of partial differential equations; polarized abelian threefolds; polarized abelian variety; complex multiplication; number of cusps of a Picard surface; class number; CM fields B. Gordon, Canonical models of Picard modular surfaces, The zeta functions of Picard modular surfaces, Centre de Recherches Mathématiques, Université de Montréal, Montréal (1992), 1-29. Modular and Shimura varieties, Global ground fields in algebraic geometry, Special surfaces, Complex multiplication and abelian varieties, Arithmetic ground fields for abelian varieties, Families, moduli, classification: algebraic theory Canonical models of Picard modular surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces modular function \(j(\tau)\); singular moduli; prime factorization of the absolute norm; modular polynomial; arithmetic of maximal orders in quaternion algebras; geometry of supersingular elliptic curves; Fourier coefficients; Eisenstein series; Hilbert modular group; local heights; Heegner points Gross, B. H.; Zagier, D. B., \textit{on singular moduli}, J. Reine Angew. Math., 355, 191-220, (1985) Modular and automorphic functions, Algebraic moduli of abelian varieties, classification, Special algebraic curves and curves of low genus, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Quaternion and other division algebras: arithmetic, zeta functions On singular moduli

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central points; model class of fields; valuation; function field Bröcker, L.; Schülting, H. W.: Valuation theory from the geometrical point of view. J. reine angew. Math. 365, 12-32 (1986) Model theory of fields, Valued fields, Algebraic functions and function fields in algebraic geometry, Model-theoretic algebra Valuations of function fields from the geometrical point of view

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central extensions of Lie algebras; conformal groups; Witt algebra; conformal field theories; central extensions of groups; two-dimensional conformal field theory; Virasoro algebra; conformal symmetries in dimension two; representation; Verma modules; Kac determinant; diffeomorphism group of the circle; bosonic string theory; Verlinde formula; fusion rule; dimension formula; spaces of generalized theta functions; moduli spaces of vector bundles; compact Riemann surfaces; bibliography Schottenloher, M.: A mathematical introduction to conformal field theory. (1997) Virasoro and related algebras, Vector bundles on curves and their moduli, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to quantum theory, Axiomatic quantum field theory; operator algebras, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations A mathematical introduction to conformal field theory. Based on a series of lectures given at the Mathematisches Institut der Universität Hamburg

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces moduli of curves; surfaces over number fields Arithmetic ground fields for surfaces or higher-dimensional varieties, Families, moduli of curves (algebraic), Coverings of curves, fundamental group Covers of moduli surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces conic bundle surfaces; Henselian fields; quaternion algebras Finite-dimensional division rings, Forms over real fields, Quadratic forms over general fields, Skew fields, division rings, Algebraic functions and function fields in algebraic geometry, Rational and ruled surfaces Note on conic bundles over Henselian discrete valued fields with real closed residue field.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces curves over a finite field; curves with many points; graphs; towers of function fields; zeta functions ] Emmanuel Hallouin and Marc Perret, From Hodge index theorem to the number of points of curves over finite fields, arXiv:1409.2357v1, 2014. Curves over finite and local fields, Rational points, Finite ground fields in algebraic geometry, Singularities of curves, local rings, Paths and cycles, Graphs and linear algebra (matrices, eigenvalues, etc.) Recursive towers of curves over finite fields using graph theory

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces actions of groups; linear algebra; topological groups; endomorphisms; Grassmannians; echelon matrices; groups preserving a bilinear form; quaternion fields; algebraic combinatorics; Lie groups; Platonic solids; topics from the projective plane; orthogonal groups; unitary groups; symplectic groups; Young tableaux; algebraic geometry; algebraic curves; surfaces configurations; special varieties; graphes; projective line; conics; representation theory; McKay correspondance Ph. Caldero, J. Germoni, \textit{Histoires Hédonistes de Groupes et de Géométries [Hedonistic Histories of Groups and Geometries].} Vol. 2, Calvage et Mounet, Paris, 2015. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, Affine analytic geometry, Projective analytic geometry, Geometry of classical groups, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Euclidean analytic geometry, Analytic geometry with other transformation groups, General theory of linear incidence geometry and projective geometries, Representations of finite symmetric groups, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Curves in algebraic geometry, Singularities of curves, local rings Hedonistic histories of groups and geometries. Vol. 2

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces existence of isometry-dual flags of codes; two-point algebraic geometry codes; isometry-dual property; two-point codes over function fields Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry The isometry-dual property in flags of two-point algebraic geometry codes

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function field; constant reduction; valuation prolongation; geometric family; Skolem property; principal family; birational characterization of arithmetic surfaces B. Green, Geometric families of constant reductions and the Skolem property, Trans. Amer. Math. Soc. 350 (1998), no. 4, 1379-1393. Valued fields, Arithmetic theory of algebraic function fields, Arithmetic ground fields for surfaces or higher-dimensional varieties, Global ground fields in algebraic geometry Geometric families of constant reductions and the Skolem property

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic curves and algebraic surfaces; resolution of singularities; local rings; valuation theory; Cohen-Macaulay rings; differential modules; analytic algebras Kiyek, K.; Vicente, J. L., Resolution of curve and surface singularities in characteristic zero, (2004), Kluwer Dordrecht Research exposition (monographs, survey articles) pertaining to algebraic geometry, Valuations and their generalizations for commutative rings, Singularities in algebraic geometry, Singularities of curves, local rings, Singularities of surfaces or higher-dimensional varieties, Analytic algebras and generalizations, preparation theorems, Sheaves of differential operators and their modules, \(D\)-modules, Global theory and resolution of singularities (algebro-geometric aspects) Resolution of curve and surface singularities in characteristic zero.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces genus of curves over finite fields; many rational points; maximal function fields R. Fuhrmann and F. Torres. The genus of curves over finite fields with many rational points. Manuscripta Math., 89(1) (1996), 103--106. Curves over finite and local fields, Arithmetic theory of algebraic function fields, Finite ground fields in algebraic geometry, Special algebraic curves and curves of low genus, Arithmetic ground fields for curves The genus of curves over finite fields with many rational points

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; Galois theory of function fields; Kummer theory; valuations; flag functions F.\ A. Bogomolov and Y. Tschinkel, Commuting elements of Galois groups of function fields, Motives, polylogarithms and Hodge theory. Part I (Irvine 1998), Int. Press Lect. Ser. 3, International Press, Somerville (2002), 75-120. Arithmetic theory of algebraic function fields, Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Configurations and arrangements of linear subspaces Commuting elements in Galois groups of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite field; tensor rank of multiplication; function field Number-theoretic algorithms; complexity, Analysis of algorithms and problem complexity, Algebraic functions and function fields in algebraic geometry, Finite fields (field-theoretic aspects) On the tensor rank of multiplication in finite extensions of finite fields and related issues in algebraic geometry

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces nonexistence of certain level structures on abelian varieties; g-Siegel upper-half plane; complex function field; compactifications of bounded symmetric domains A. M. Nadel, ``The nonexistence of certain level structures on abelian varieties over complex function fields'', Ann. of Math. (2)129 (1989) no. 1, p. 161-178 Analytic theory of abelian varieties; abelian integrals and differentials, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) The nonexistence of certain level structures on abelian varieties over complex function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Hasse-Schmidt derivations; vertex operators on exterior algebras; representation of Lie algebras of matrices; bosonic and fermionic representations by Date-Jimbo-Kashiwara-Miwa; symmetric functions Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Vertex operators; vertex operator algebras and related structures, Exterior algebra, Grassmann algebras, Symmetric functions and generalizations On the vertex operator representation of Lie algebras of matrices

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces cohomological dimension of fields; \(C_i\) property; Milnor K-theory; number fields; function fields Forms of degree higher than two, Hilbertian fields; Hilbert's irreducibility theorem, Field arithmetic, Other nonalgebraically closed ground fields in algebraic geometry, Higher symbols, Milnor \(K\)-theory, \(K\)-theory in number theory On a conjecture of Kato and Kuzumaki

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces resolution of cusp singularities; Shintani decomposition; totally real cubic number fields; Hilbert modular variety; family of cubics; evaluation of zeta-function DOI: 10.1007/BF01359864 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Singularities of surfaces or higher-dimensional varieties, Special surfaces, \(3\)-folds, Totally real fields, Cubic and quartic extensions, Global ground fields in algebraic geometry On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite field; towers of algebraic function fields Arithmetic theory of algebraic function fields, Class field theory, Algebraic functions and function fields in algebraic geometry, Rational points A note on towers of function fields over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces skew fields; central simple algebras; quaternion algebras; Galois extension; quadratic forms; Merkur'ev theorem; Brauer group Division rings and semisimple Artin rings, Grothendieck groups, \(K\)-theory, etc., Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Finite rings and finite-dimensional associative algebras, Brauer groups of schemes The Merkuryev-Suslin theorem

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces stable reduction of curves; completely valued fields; ultrametric valuation; topological function field; topological genus; inequalities Non-Archimedean valued fields, Algebraic functions and function fields in algebraic geometry Genre topologique des corps valués. (Topological genus of valued fields)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces ruled function fields; Zariski problem; finitely generated extensions; automorphisms James K. Deveney, Automorphism groups of ruled function fields and a problem of Zariski, Proc. Amer. Math. Soc. 90 (1984), no. 2, 178 -- 180. Transcendental field extensions, Algebraic functions and function fields in algebraic geometry Automorphism groups of ruled function fields and a problem of Zariski

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces non-vanishing of \(L\)-functions; twisted \(L\)-functions of elliptic curves; function fields; elliptic curve rank in extensions Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Arithmetic ground fields for curves On the vanishing of twisted \(L\)-functions of elliptic curves over rational function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces differential graded categories; triangulated categories; derived noncommutative schemes; finite-dimensional algebras; geometric realizations; noncommutative algebraic geometry; quasi-coherent sheaves; homological algebra; perfect complexes; unbounded derived category; enough injectives; classical generator; homotopy category; enhanced category; noncommutative scheme; noncommutative derived scheme; compactification; resolution of singularities; Serre functor; geometric realization; pure geometric realization; phantoms; quasi-phantoms; Krull-Schmidt partners Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Noncommutative algebraic geometry, Differential graded algebras and applications (associative algebraic aspects), Derived categories, triangulated categories, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Chain complexes (category-theoretic aspects), dg categories Derived noncommutative schemes, geometric realizations, and finite dimensional algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces moments of quadratic Dirichlet; \(L\)-functions; ratios of \(L\)-functions; function fields; random matrix theory; hyperelliptic curves Andrade, J. C.; Keating, J. P., Conjectures for the integral moments and ratios of \textit{L}-functions over function fields, J. Number Theory, 142, 102-148, (2014) Curves over finite and local fields, Relations with random matrices, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Conjectures for the integral moments and ratios of \(L\)-functions over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces quasi-split symmetric pairs; Schur algebras; quantization of \(q\)-Schur algebras; isotropic flags; Chevalley generators; fixed point subalgebra Universal enveloping (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Schur and \(q\)-Schur algebras Quasi-split symmetric pairs of \(U(\mathfrak{gl}_N)\) and their Schur algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Brauer group of the rational function fields Brauer groups of schemes, Galois cohomology, Arithmetic theory of algebraic function fields Groupe de Brauer des corps de fractions rationnelles à coefficients complexes

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields; curves of genus greater than 1; finite number of points defined over the ground field Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic theory of algebraic function fields, Rational points Diophantine equations over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Tate-Shafarevich group; local-global principle; patching; central simple algebras; Brauer group; arithmetic duality; function fields; higher-dimensional local fields Izquierdo, Diego Principe local-global pour les corps de fonctions sur des corps locaux supérieurs I \textit{J.~Number Theory}157 (2015) 250--270 Math Reviews MR3373241 Galois cohomology of linear algebraic groups, Galois cohomology, Other nonalgebraically closed ground fields in algebraic geometry, Galois cohomology Local-global principle for function fields over higher local fields. I

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces reciprocity law for surfaces over finite fields; group of degree 0 zero- cycles; rational equivalence; abelian geometric fundamental group; unramified class field theory; K-theory; Chow groups Jean-Louis Colliot-Thélène & Wayne Raskind, ``On the reciprocity law for surfaces over finite fields'', J. Fac. Sci. Univ. Tokyo Sect. IA Math.33 (1986) no. 2, p. 283-294 Finite ground fields in algebraic geometry, Coverings in algebraic geometry, Algebraic cycles, Parametrization (Chow and Hilbert schemes), Homotopy theory and fundamental groups in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Arithmetic theory of algebraic function fields On the reciprocity law for surfaces over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Bogomolov conjecture; curves of higher genus; function fields; metric graphs Faber, X. W. C., The geometric Bogomolov conjecture for curves of small genus, Experiment. Math., 1058-6458, 18, 3, 347\textendash 367 pp., (2009) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic varieties and schemes; Arakelov theory; heights, Heights The geometric Bogomolov conjecture for curves of small genus

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces generalized Cartan matrix; Kac-Moody Lie algebras; Kac-Moody groups; projective normality of Schubert varieties; tensor product of two G- modules O. Mathieu : Construction du groupe associé aux algèbres de Kac-Moody . Comptes Rendus 306 série I (1988) 227-230. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Group schemes, Infinite-dimensional Lie groups and their Lie algebras: general properties Construction du groupe de Kac-Moody et applications. (Construction of the Kac-Moody group with applications)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Grassmann manifold; Grassmannian; product of Grassmannians; Grassmann quasi-Newton; Grassmann BFGS; Grassmann limited memory BFGS; multilinear rank; symmetric multilinear rank; tensor; symmetric tensor; approximations; Broyden-Fletcher-Shanno-Goldfarb (BFGS) B. Savas and L.-H. Lim, \textit{Quasi-Newton methods on Grassmannians and multilinear approximations of tensors}, SIAM J. Sci. Comput., 32 (2010), pp. 3352--3393, . Numerical mathematical programming methods, Methods of quasi-Newton type, Multilinear algebra, tensor calculus, Grassmannians, Schubert varieties, flag manifolds, Nonlinear programming, Differential geometric aspects in vector and tensor analysis Quasi-Newton methods on Grassmannians and multilinear approximations of tensors

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces specialization of Galois extensions; function fields; Chebotarev property; Hilbert's irreducibility theorem; local and global fields Checcoli, S.; Dèbes, P.: Tchebotarev theorems for function fields. (2013) Arithmetic theory of algebraic function fields, Separable extensions, Galois theory, Hilbertian fields; Hilbert's irreducibility theorem, Field arithmetic, Arithmetic problems in algebraic geometry; Diophantine geometry Tchebotarev theorems for function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic functions of one variable; algebraic function fields; arbitrary field of constants Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Fields of algebraic functions of one variable over an arbitrary field of constants

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces gamma-function; zeta-function; matrix integrals; symmetric spaces; flag spaces; lattices; distribution of eigenvalues Gamma, beta and polygamma functions, Differential geometry of symmetric spaces, \(p\)-adic theory, local fields, Grassmannians, Schubert varieties, flag manifolds Matrix beta-integrals: an overview

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces defect of the valued function fields; genus; ramification index Michel Matignon, Genre et genre résiduel des corps de fonctions valués, Manuscripta Math. 58 (1987), no. 1-2, 179 -- 214 (French, with English summary). Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Ramification problems in algebraic geometry, Valued fields Genre et genre residuel des corps de fonctions valués. (Genus and residual genus of valued function fields)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces rings of differential operators; genus; noncommutative surfaces; birational invariants Noncommutative algebraic geometry, Rings of differential operators (associative algebraic aspects), Rings arising from noncommutative algebraic geometry Embeddings of quotient division algebras of rings of differential operators

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces arithmetic properties of hyperelliptic function fields; minimum distance of geometric codes; hyperelliptic curves Xing, C. -P.: Hyperelliptic function fields and codes. J. pure appl. Algebra 74, 109-118 (1991) Geometric methods (including applications of algebraic geometry) applied to coding theory, Algebraic coding theory; cryptography (number-theoretic aspects), Algebraic functions and function fields in algebraic geometry Hyperelliptic function fields and codes

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Brauer groups; Hasse principle; function fields of genus 1 Brauer groups of schemes, Algebraic functions and function fields in algebraic geometry, Special algebraic curves and curves of low genus A method of computing the constant field obstruction to the Hasse principle for the Brauer groups of genus one curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces densities of discriminants of cubic fields; 3-class-number of quadratic fields; binary cubic forms; adelization; zeta-functions; function field; Dedekind's zeta-function Iwasawa theory, Arithmetic theory of algebraic function fields, Quadratic forms over global rings and fields, Density theorems, Asymptotic results on counting functions for algebraic and topological structures, Cubic and quartic extensions, Global ground fields in algebraic geometry, Zeta functions and \(L\)-functions of number fields, Algebraic functions and function fields in algebraic geometry Density of discriminants of cubic extensions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces finite fields; towers of function fields; congruence zeta functions DOI: 10.3836/tjm/1202136690 Arithmetic theory of algebraic function fields, Finite ground fields in algebraic geometry, Algebraic functions and function fields in algebraic geometry A note on optimal towers over finite fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields; algebraic curves; distributions of values DOI: 10.1090/S0002-9947-06-04018-9 Algebraic functions and function fields in algebraic geometry, Value distribution of meromorphic functions of one complex variable, Nevanlinna theory, Riemann surfaces; Weierstrass points; gap sequences Unique range sets and uniqueness polynomials for algebraic curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Belyı curves; Beauville surfaces; fields of definition; Higgs fields; rigidity; Shimura varieties; spreads Arithmetic aspects of modular and Shimura varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects), Modular and Shimura varieties Rigidity of spreadings and fields of definition

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function theory of surfaces; principal congruence groups; Hecke subgroups; characteristic classes; punctures; Riemann surface; theta constant identities; conformal mappings Holomorphic modular forms of integral weight, Theta series; Weil representation; theta correspondences, Compact Riemann surfaces and uniformization, Riemann surfaces; Weierstrass points; gap sequences Automorphic forms for subgroups of the modular group. II: Groups containing congruence subgroups

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces homological mirror symmetry; wrapped Fukaya category; symmetric products of surfaces; matrix factorizations Mirror symmetry (algebro-geometric aspects), Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category Homological mirror symmetry for the symmetric squares of punctured spheres

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces quantum complete intersections; vanishing of cohomology; symmetric algebras; graded modules Bergh, PA, Ext-symmetry over quantum complete intersections, Arch. Math. (Basel), 92, 566-573, (2009) Homological functors on modules (Tor, Ext, etc.) in associative algebras, Graded rings and modules (associative rings and algebras), Complete intersections Ext-symmetry over quantum complete intersections.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces \(K3\) surface, double cover family of \(K3\) surfaces, period, mirror map, \(K3\) lambda function, hypergeometric system, mirror symmetry \(K3\) surfaces and Enriques surfaces, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Families, moduli, classification: algebraic theory, Mirror symmetry (algebro-geometric aspects) \(K3\) surfaces from configurations of six lines in \(\mathbb{P}^2\) and mirror symmetry. II: \(\lambda_{K3}\)-functions

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Siegel modular forms; \(K3\) surfaces; Picard rank sixteen; van Geemen-Sarti \(K3\) surfaces; double sextic \(K3\) surfaces; F-theory/heteotic string duality in eight dimensions.; van Geemen-Sarti \(K3\) surfaces; moduli space of six lines; bounded symmetric domains of type IV \(K3\) surfaces and Enriques surfaces, Modular and automorphic functions, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Relationships between surfaces, higher-dimensional varieties, and physics Six line configurations and string dualities

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Ulm invariants; Brauer group of algebraic function fields over global fields Fein, B.; Schacher, M.: Brauer groups of algebraic function fields. J. algebra 103, 454-465 (1986) Arithmetic theory of algebraic function fields, Galois cohomology, Brauer groups of schemes Brauer groups of algebraic function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces towers of function fields; rational places; genus of a function field; automorphisms of function fields; \(p\)-rank Arithmetic theory of algebraic function fields, Curves over finite and local fields, Applications to coding theory and cryptography of arithmetic geometry, Algebraic functions and function fields in algebraic geometry Asymptotically good towers of function fields with small \(p\)-rank

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces central simple algebras; strong approximation property; commutator subgroups; rational function fields; global fields; Brauer groups Infinite-dimensional and general division rings, Arithmetic theory of algebraic function fields, Galois cohomology, Brauer groups of schemes, Adèle rings and groups Strong approximation theorem for division algebras over \(\mathbb{R}(X)\)

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Riemann surfaces; algebraic curves; automorphisms; fields of moduli Hidalgo, R. A., Erratum: non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals, Arch. Math., 98, pp., (2012) Compact Riemann surfaces and uniformization, Automorphisms of curves Erratum to: Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces reduced Whitehead groups; Tannaka-Artin problem; patching; \(\mathrm{SK}_1\); function fields of \(p\)-adic curves Finite-dimensional division rings, Brauer groups (algebraic aspects), Arithmetic theory of algebraic function fields, Arithmetic ground fields for curves, Algebraic functions and function fields in algebraic geometry Reduced Whitehead groups of prime exponent algebras over \(p\)-adic curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces automorphic forms on function fields; automorphic cuspidal module; filtration on the moduli stack of shtukas; absolute values of the complex Hecke eigenvalues; full trace formula; residual spectrum Representation-theoretic methods; automorphic representations over local and global fields, Formal groups, \(p\)-divisible groups On the Ramanujan-Petersson conjecture over function fields. II: Spectral study

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces secant variety; Segre variety; Veronese embedding; products of projective spaces; defectivity; tensor rank; partially symmetric tensor; Grassmann defectivity; Segre-Veronese embeddings M. V. Catalisano, A. V. Geramita, and A. Gimigliano, Higher secant varieties of Segre-Veronese varieties, Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 81 -- 107. Adam Van Tuyl, An appendix to a paper of M. V. Catalisano, A. V. Geramita and A. Gimigliano. The Hilbert function of generic sets of 2-fat points in \Bbb P\textonesuperior \times \Bbb P\textonesuperior : ''Higher secant varieties of Segre-Veronese varieties'' [in Projective varieties with unexpected properties, 81 -- 107, Walter de Gruyter GmbH & Co. KG, Berlin, 2005; MR2202248], Projective varieties with unexpected properties, Walter de Gruyter, Berlin, 2005, pp. 109 -- 112. Projective techniques in algebraic geometry, Determinantal varieties, Multilinear algebra, tensor calculus Higher secant varieties of Segre-Veronese varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces families of low dimensions; deformations of complex structures; analytic moduli problem; compact surfaces; removable singularities; fibres; local triviality; degeneracy loci; ruled surfaces; Hopf surfaces; Inoue surfaces; structure-jumping in higher codimension; reduction of coefficients; rank-3 holomorphic vector bundles on \(\mathbb C\mathbb P^n\) Complex-analytic moduli problems, Deformations of complex structures, Compact complex surfaces, Families, moduli of curves (analytic) Remark on the article ``Singular loci in holomorphic families''

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Riemann surface; symmetry of Riemann surface; asymmetric Riemann surface; pseudo-symmetric Riemann surface; Fuchsian groups; NEC groups; fixed points on Riemann surfaces; fixed point free actions on Riemann surfaces Automorphisms of curves, Riemann surfaces Fixed points on asymmetric Riemann surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function field; rational place; Weierstrass semigroup; tower of function fields Geil O., Matsumoto R.: Bounding the number of \(\mathbb{F}_q\)-rational places in algebraic function fields using Weierstrass semigroups. J. Pure Appl. Algebra \textbf{213}(6), 1152-1156 (2009). Finite ground fields in algebraic geometry, Curves over finite and local fields, Algebraic functions and function fields in algebraic geometry, Arithmetic ground fields for curves Bounding the number of \(\mathbb F_q\)-rational places in algebraic function fields using Weierstrass semigroups

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic surfaces; Kähler manifolds; moduli; deformations; topological methods; fibrations; Kodaira fibrations; Chern slope; automorphisms; uniformization; projective classifying spaces; monodromy; fundamental groups; variation of Hodge structure; absolute Galois group; locally symmetric varieties Pencils, nets, webs in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Fine and coarse moduli spaces, Coverings in algebraic geometry, Modular and Shimura varieties, Coverings of curves, fundamental group, Surfaces of general type, Automorphisms of surfaces and higher-dimensional varieties, Kähler-Einstein manifolds, Uniformization of complex manifolds, Transcendental methods of algebraic geometry (complex-analytic aspects), Topological aspects of complex manifolds, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects), General theory of automorphic functions of several complex variables, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Period matrices, variation of Hodge structure; degenerations, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) Kodaira fibrations and beyond: methods for moduli theory

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Klein mapping; line congruencies; ruled surfaces; polygonal schema of a surface; homology groups Line geometries and their generalizations, Grassmannians, Schubert varieties, flag manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Differential line geometry, Computer science aspects of computer-aided design On topological properties of some line congruencies

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces function fields of \(p\)-adic curves; classical groups; projective homogeneous spaces; local-global principle; unitary groups Galois cohomology of linear algebraic groups, Curves over finite and local fields, Rational points, Local ground fields in algebraic geometry Local-global principle for classical groups over function fields of \(p\)-adic curves

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces homogeneous Poisson-commutative subalgebras; Mishchenko-Fomenko algebras; Gelfand-Tsetlin algebra; symmetric algebra of reductive Lie algebras; completeness on co-adjoint orbits; nilpotent orbits; integrable systems; moment map; coisotropic actions Poisson algebras, Group actions on varieties or schemes (quotients), Coadjoint orbits; nilpotent varieties, Simple, semisimple, reductive (super)algebras, Semisimple Lie groups and their representations Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Parshin's conjecture; Miyaoka-Yau inequality; surfaces of general type; fields of positive characteristic Jang, J, Generically ordinary fibrations and a counterexample to parshin's conjecture, Mich. Math. J., 59, 169-178, (2010) Arithmetic ground fields for surfaces or higher-dimensional varieties, Varieties over finite and local fields Generically ordinary fibrations and a counterexample to Parshin's conjecture

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces arithmetic theory of algebraic function fields Lettl, G, Thue equations over algebraic function fields, Acta Arith., 117, 107-123, (2005) Thue-Mahler equations, Arithmetic theory of algebraic function fields, Rational points, Cubic and quartic Diophantine equations Thue equations over algebraic function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Weil divisor; structure of normal surfaces admitting a minimal ruled fibration; ampleness criterion; Nagata inequality for the self- intersection numbers; classification Sakai, F.: Ruled fibrations on normal surfaces. J. Math. Soc. Jpn. 40(2), 249-269 (1988) Families, moduli, classification: algebraic theory, Rational and ruled surfaces, Divisors, linear systems, invertible sheaves Ruled fibrations on normal surfaces

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces algebraic function fields of genus one; real-closed field; J-invariant Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Real algebraic and real-analytic geometry, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) An isomorphism theorem for algebraic function fields of genus one over real-closed fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces hyperelliptic curves; local fields; admissible pairing; self-intersection of the relative dualising sheaf; symmetric roots Robin de Jong, Symmetric roots and admissible pairing, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4263 -- 4283. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic varieties and schemes; Arakelov theory; heights Symmetric roots and admissible pairing

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces genus of curve; stable reduction of curve; topological function field; complete non-Archimedean valued fields; topological genus Valued fields, Transcendental field extensions, Algebraic functions and function fields in algebraic geometry Genre topologique de corps valués

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces symmetric tensor rank; additive decomposition of polynomials; Waring decomposition; identifiability Secant varieties, tensor rank, varieties of sums of powers, Projective techniques in algebraic geometry, Multilinear algebra, tensor calculus, Vector and tensor algebra, theory of invariants, Configurations and arrangements of linear subspaces An effective criterion for the additive decompositions of forms

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces embeddings of central simple algebras; exponent; index; degree; invariants; Brauer groups; number fields DOI: 10.1016/0021-8693(90)90277-U Finite-dimensional division rings, Brauer groups of schemes Minimal embeddings of central simple algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces isotriviality; effective Mordell; semiabelian variety; positive characteristic; survey of diophantine geometry; bounding the heights of rational points on curves over function fields; semiabelian varieties; Roth's theorem Voloch, José Felipe, Diophantine geometry in characteristic \(p\): a survey.Arithmetic geometry, Cortona, 1994, Sympos. Math., XXXVII, 260-278, (1997), Cambridge Univ. Press, Cambridge Rational points, Local ground fields in algebraic geometry, Arithmetic algebraic geometry (Diophantine geometry), Arithmetic ground fields for curves Diophantine geometry in characteristic \(p\): A survey

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces numerical effective bundle; higher dimensional analogue of Mordell's finiteness conjecture over function fields; nef Moduli, classification: analytic theory; relations with modular forms, Rational and birational maps, Algebraic functions and function fields in algebraic geometry The Mordell-Bombieri-Noguchi conjecture over function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces K3-surfaces; integral symmetric bilinear form; involution; system of invariants; genus; isomorphism classes; real nonsingular projective algebraic variety; new invariants of real algebraic surfaces; relations; standard invariants; Euler characteristic Nikulin V.V. (1984). Involutions of integral quadratic formsand their applications to real algebraic geometry. Math. USSR Izv. 22: 99--172 General binary quadratic forms, Quadratic forms over global rings and fields, Surfaces and higher-dimensional varieties, Real algebraic and real-analytic geometry Involutions of integral quadratic forms and their applications to real algebraic geometry

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Mordell conjecture for function fields; theorem of the kernel doi:10.2307/2374831 Rational points, Arithmetic theory of algebraic function fields, History of algebraic geometry A note on Manin's theorem of the kernel

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Cherednik algebras; Morita equivalences; spherical algebras; Hecke algebras; Coxeter groups; symmetric groups; root systems; algebras of differential operators; algebras of quasi-invariants Berest Y., Etingof P., Ginzburg V.: Morita equivalence of Cherednik algebras. J. Reine Angew. Math. 568, 81--98 (2004) Rings of differential operators (associative algebraic aspects), Module categories in associative algebras, Noncommutative algebraic geometry, Simple, semisimple, reductive (super)algebras, Hecke algebras and their representations, Reflection and Coxeter groups (group-theoretic aspects), Representations of finite symmetric groups, Geometric invariant theory Morita equivalence of Cherednik algebras.

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces deformation theory; Deligne conjecture; quantum algebra; rings and algebras; category theory; algebraic topology; algebraic geometry; operads; geometry of configuration spaces of points on surfaces; polynomial functors; minimal operad Kontsevich, M.; Soibelman, Y., Deformations of algebras over operads and the Deligne conjecture, (Proceedings of the Moshé Flato Conference. Proceedings of the Moshé Flato Conference, Math. Phys. Stud., vol. 21, (2000), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 255-307 Curves in algebraic geometry, Deformations of associative rings Deformations of algebras over operads and the Deligne conjecture

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces elliptic divisibility sequences; elliptic surfaces; primitive divisors; function fields; constant \(j\)-invariant Elliptic curves over global fields, Fibonacci and Lucas numbers and polynomials and generalizations, Elliptic curves, Elliptic curves over local fields, Special sequences and polynomials Primitive divisors of elliptic divisibility sequences over function fields with constant \(j\)-invariant

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Weierstrass semigroup; asymptotically good tower of function fields Pellikaan R., Stichtenoth H., Torres F. (1998). Weierstrass semigroups in an asymptotically good tower of function fields. Finite Fields Appl 4(4):381--392 Arithmetic theory of algebraic function fields, Curves over finite and local fields, Geometric methods (including applications of algebraic geometry) applied to coding theory, Arithmetic ground fields for curves, Riemann surfaces; Weierstrass points; gap sequences Weierstrass semigroups in an asymptotically good tower of function fields

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Chow groups; tensor triangulated geometry; maximal orders; derived categories of sheaves; noncommutative algebraic geometry Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) Relative tensor triangular Chow groups for coherent algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Newman's conjecture; zeros of the Riemann zeta function; \(L\)-functions; function fields; random matrix theory; Sato-Tate conjecture Andrade, J.; Chang, A.; Miller, S. J.: Newman's conjecture in various settings. J. number theory 144, 70-91 (2013) Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\), Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses, Relations with random matrices, Analytic computations, Evaluation of number-theoretic constants, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Newman's conjecture in various settings

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces zeta function of a number field; arithmetically equivalent number fields; Gassmann triple; permutation representations; integral representations; idele class groups; algebraic curves; Jacobians Zeta functions and \(L\)-functions of number fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Jacobians, Prym varieties, Group rings of finite groups and their modules (group-theoretic aspects) A refined notion of arithmetically equivalent number fields, and curves with isomorphic Jacobians

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces moduli of punctured Riemann surfaces; WZWN-theory; conformal blocks; Krichever-Novikov algebras; higher genus Knizhnik-Zamolodchikov equations; Sugawara construction; higher genus Riemann surfaces M. Schlichenmaier and O. K. Sheinman, ''The Wess-Zumino-Witten-Novikov Theory, Knizhnik-Zamolodchikov Equations, and Krichever-Novikov Algebras,'' Russ. Math. Surveys 54(1), 213--249 (1999). Virasoro and related algebras, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Applications of Lie (super)algebras to physics, etc., Relationships between algebraic curves and physics Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces Frobenius endomorphisms; curves over finite fields; projective curve of genus 5; zeta function Lauter K., Proceedings of the American Mathematical Society 128 (2) pp 369-- (2000) Curves over finite and local fields, Rational points, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry Non-existence of a curve over \(\mathbb F_3\) of genus 5 with 14 rational points

noncommutative symmetric algebras; bimodules; tensor algebras; skew fields of frractions; function fields; noncommutative ruled surfaces Patrick D., J. Algebra 233 pp 16-- (2000) Rings arising from noncommutative algebraic geometry, Endomorphism rings; matrix rings, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Rational and ruled surfaces Noncommutative symmetric algebras of two-sided vector spaces theory of algebraic curves; coding theory; Riemann-Roch theorem; function fields; differentials; Hasse-Weil theorem; geometric Goppa codes; trace codes H. Stichtenoth, Algebraic Function Fields and Codes, Second edn, (Springer-Verlag, Berlin Heidelberg, 2009). Zbl0816.14011 MR2464941 Algebraic functions and function fields in algebraic geometry, Arithmetic theory of algebraic function fields, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory, Research exposition (monographs, survey articles) pertaining to information and communication theory, Geometric methods (including applications of algebraic geometry) applied to coding theory, Cyclic codes Algebraic function fields and codes