Showing papers on "Field (mathematics) published in 1998"
Book•
10 Dec 1998
TL;DR: L-subsets and L-subgroups of Abelian groups as discussed by the authors, which are a generalization of group L -subalgebras, and are the basis of the L-ideals structure.
Abstract: L-subsets and L-subgroups L-subgroups of Abelian groups L-submodules L-subrings and L-ideals L-subfields structure of L-subrings and L-ideals algebraic L-varieties and intersection equations L-subspaces Galois theory and group L-subalgebras.
235 citations
Journal Article•
TL;DR: A class of Galois field used to achieve fast finite field arithmetic which is called an Optimal Extension Field (OEF) is introduced, well suited for implementation of public-key cryptosystems based on elliptic and hyperelliptic curves.
Abstract: This contribution introduces a class of Galois field used to achieve fast finite field arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of public-key cryptosystems based on elliptic and hyperelliptic curves. Whereas previous reported optimizations focus on finite fields of the form GF(p) and GF(2 m ), an OEF is the class of fields GF(p m ), for p a prime of special form and m a positive integer. Modern RISC workstation processors are optimized to perform integer arithmetic on integers of size up to the word size of the processor. Our construction employs well-known techniques for fast finite field arithmetic which fully exploit the fast integer arithmetic found on these processors. In this paper, we describe our methods to perform the arithmetic in an OEF and the methods to construct OEFs. We provide a list of OEFs tailored for processors with 8, 16, 32, and 64 bit word sizes. We report on our application of this approach to construction of elliptic curve cryptosystems and demonstrate a substantial performance improvement over all previous reported software implementations of Galois field arithmetic for elliptic curves.
212 citations
TL;DR: In this article, a spin-triplet odd-parity superconductor with a nonunitary spin-parallel spin pairing was shown to have a two-component ρ-vector with a superconducting transition temperature of 28 mK, regardless of major crystal orientations.
Abstract: ${}^{195}\mathrm{Pt}$ Knight shift (KS) measurements covering the superconducting multiple phases for major field ( $H$) orientations have been carried out on the high-quality single crystal ${\mathrm{UPt}}_{3}$. For $Hg5$ kOe, the KS does not change below the superconducting transition temperature ${T}_{c}$ down to 28 mK, regardless of major crystal orientations, which provides evidence that the odd-parity superconductivity with the parallel spin pairing is realized. By contrast, the KS decreases below ${T}_{c}$ for ${H}_{b}\ensuremath{\parallel}b$ axis and ${H}_{b}l5$ kOe and for ${H}_{c}\ensuremath{\parallel}c$ axis and ${H}_{c}l2.3$ kOe, whereas the KS for ${H}_{a}\ensuremath{\parallel}a$ axis is $T$ independent across ${T}_{c}$ down to ${H}_{a}\ensuremath{\sim}1.764$ kOe. These novel findings entitle ${\mathrm{UPt}}_{3}$ as the first spin-triplet odd-parity superconductor including a nonunitary pairing characterized by the two-component $\mathbf{d}$ vector like ${\mathbf{d}}_{b}+i{\mathbf{d}}_{c}$ at low $T$ and low $H$.
190 citations
Posted Content•
TL;DR: In this article, the authors studied the gravitational descendents of Gromov-Witten invariants for general projective manifolds, applying the Behrend-Fantechi construction of the virtual fundamental classes.
Abstract: In Part 1 of this paper, we study gravitational descendents of Gromov-Witten invariants for general projective manifolds, applying the Behrend-Fantechi construction of the virtual fundamental classes.
In Part 2, we calculate the topological recursion relations in genus 2. There are two of these, one for the second descendent of a field, and one for the first descendents of two fields. The proof uses the results of Part 1 together with a thorough study of intersection theory on the moduli space $\bar{M}_{2,2}$.
158 citations
TL;DR: In this article, a model complete and o-minimal expansion of the field of real numbers was constructed, in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn 1 is definable.
Abstract: We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn 1 is definable. This expansion is polynomially bounded.
123 citations
TL;DR: In this article, the relation between the non-cuspidal irreducible smooth representations of the F -points of a reductive connected algebraic group over F over an R-vector space, and the simple modules of affine Hecke algebras with coefficients in R, where R is an algebraically closed field of positive characteristic = p, and F is a local non archimedean field, of finite residual field with q elements and characteristic p.
Abstract: This article concerns the relation between the non cuspidal irreducible smooth representations of the F -points of a reductive connected algebraic group over F over an R-vector space, and the simple modules of affine Hecke algebras with coefficients in R, where R is an algebraically closed field of positive characteristic = p, and F is a local non archimedean field, of finite residual field with q elements and characteristic p. When G = GL(n, F ): 1) We describe a set of unipotent irreducible R-representations of GL(n, F ) (i.e. subquotients of parabolically induced representations from unramified characters of a Borel subgroup), classified by the Deligne-Langlands R-parameters. In particular, we get a set of superunipotent irreducible R-representations of GL(n, F ) (i.e with an Iwahori fixed non zero vector), classified by the Deligne-Langlands R-parameters without cycle (V.6). They are in bijection with a set of simple modules for the affine Hecke R-algebras of type A and parameter the image of q in R (which is invertible in the prime field of R hence a root of unity). 2) We reduce the classification of all irreducible R-representations of GL(n, F ) to the classification of the unipotent irreducible representations and of the supercuspidal irreducible representations, or to the classification of the superunipotent representations and of the cuspidal representations.
123 citations
TL;DR: In this article, the authors give a qualitative classification of all compact subgroups Γ ⊂ GL n (F ), where F is a local field and n is arbitrary, up to finite index and a finite number of abelian subquotients.
114 citations
TL;DR: Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of positive characteristic as mentioned in this paper, and it is sufficient and sufficient to consider semi-simplity over such a field and then to use lifting theorem 2.1 to prove it.
Abstract: Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky's 5th conjecture on the order of the antipode of A. These results have already been proved over a field of characteristic zero, so in a sense we demonstrate that it is sufficient to consider semisimple Hopf algebras over such a field (they are also cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for semisimple and cosemisimple Hopf algebras over a field of positive characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments of deformation theory from positive to zero characteristic. The key ingredient of the proof is the theorem that the bialgebra cohomology groups of A vanish.
110 citations
01 Jan 1998
106 citations
TL;DR: The first rigorous proof of the N = 2 non-renormalization theorem was given in this article, where the absence of ultraviolet divergences beyond the one-loop level was shown.
106 citations
Posted Content•
TL;DR: A ring with an Auslander dualizing complex is a generalization of a ring with a general Auslander-Gorenstein ring as mentioned in this paper, and it is shown that many results which hold for Auslander Gromov rings also hold in the more general setting.
Abstract: A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently.
The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to non-graded algebras.
We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes.
01 Jan 1998
TL;DR: Weispfenning et al. as mentioned in this paper introduced a new decision and quantifier elimination method for the elementary formal theory of real numbers with variations and optimizations with the aim of establishing the theoretical complexity of the problem and of finding methods that are of practical importance.
Abstract: Quantifier elimination for the elementary formal theory of real numbers is a fascinating area of research at the intersection of various field of mathematics and computer science, such as mathematical logic, commutative algebra and algebraic geometry, computer algebra, computational geometry and complexity theory. Originally the method of quantifier elimination was invented (among others by Th. Skolem) in mathematical logic as a technical tool for solving the decision problem for a formalized mathematical theory. For the elementary formal theory of real numbers (or more accurately of real closed fields) such a quantifier elimination procedure was established in the 1930s by A. Tarski, using an extension of Sturm’s theorem of the 1830s for counting the number of real zeros of a univariate polynomial in a given interval. Since then an abundance of new decision and quantifier elimination methods for this theory with variations and optimizations has been published with the aim both of establishing the theoretical complexity of the problem and of finding methods that are of practical importance (see Arnon 1988a and the discussion and references in Renegar 1992a, 1992b, 1992c for a comparison of these methods). For sub-problems such as elimination of quantifiers with respect to variables, that are linearly or quadratically restricted, specialized methods have been developed with good success (see Weispfenning 1988; Loos and Weispfenning 1993; Hong 1992d; Weispfenning 1997).
TL;DR: In this paper, a characteristic-free algorithm for reducing an algebraic variety defined by the vanishing of a set of integer polynomials is described, which is used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field.
Abstract: We describe a characteristic-free algorithm for “reducing” an algebraic variety defined by the vanishing of a set of integer polynomials In very special cases, the algorithm can be used to decide whether the number of points on a variety, as the ground field varies over finite fields, is a polynomial function of the size of the field The algorithm is then used to investigate a conjecture of Kontsevich regarding the number of points on a variety associated with the set of spanning trees of any graph We also prove several theorems describing properties of a (hypothetical) minimal counterexample to the conjecture, and produce counterexamples to some related conjectures
Posted Content•
TL;DR: Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of positive characteristic as discussed by the authors, and it is sufficient and sufficient to consider semi-simplity over such a field and then to use lifting theorem 2.1 to prove it.
Abstract: Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky's 5th conjecture on the order of the antipode of A. These results have already been proved over a field of characteristic zero, so in a sense we demonstrate that it is sufficient to consider semisimple Hopf algebras over such a field (they are also cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for semisimple and cosemisimple Hopf algebras over a field of positive characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments of deformation theory from positive to zero characteristic. The key ingredient of the proof is the theorem that the bialgebra cohomology groups of A vanish.
TL;DR: In this paper, the Seshadri constant of a polarized abelian variety (A,L) is studied, which measures how much of the positivity of L can be concentrated at any given point of A. The number e(L) can be defined as the rate of growth in k of the number of jets that one can specify in the linear series |OA(kL)|.
Abstract: The purpose of this paper is to study the Seshadri constants of abelian varieties. Consider a polarized abelian variety (A,L) of dimension g over the field of complex numbers. One can associate to (A,L) a real number e(A,L), its Seshadri constant, which in effect measures how much of the positivity of L can be concentrated at any given point of A. The number e(A,L) can be defined as the rate of growth in k of the number of jets that one can specify in the linear series |OA(kL)|. Alternatively, one considers the blow-up f : X = Blx(X) −→ X of X at a point x with exceptional divisor E ⊂ X over x, and defines e(A,L) =def sup{ e ∈ R | f ∗L− eE is nef } .
TL;DR: In this article, a standard monomial theory for generic Hankel matrices was established and the symbolic and ordinary Rees algebras of It are Cohen Macaulay normal domains.
TL;DR: In this article, a generalization of Rentschler's theorem and a criterion for the existence of a slice were proposed. But they were applied to describe rank two locally nilpotent R -derivations of the polynomial ring R [ X, Y, Z ] where k is a field of characteristic zero.
Posted Content•
TL;DR: In this paper, a new height function for a variety defined over a finitely generated field over Q was proposed, and the original Raynaud's theorem (Manin-Mumford's conjecture) was recovered.
Abstract: In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott's theorem and Bogomolov's conjecture, so that we can recover the original Raynaud's theorem (Manin-Mumford's conjecture).
TL;DR: In this article, it was shown that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large, and the authors considered locally simple abelian covers of rational surfaces.
Abstract: It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.
TL;DR: In this paper, an associative algebra ℬ� R676, whose exchange properties are inferred from the scattering processes in integrable models with reflecting boundary conditions on the half line is introduced.
Abstract: Some algebraic aspects of field quantization in space-time with boundaries are discussed. We introduce an associative algebra ℬ
R
, whose exchange properties are inferred from the scattering processes in integrable models with reflecting boundary conditions on the half line. The basic properties of ℬ
R
are established and the Fock representations associated with certain involutions in ℬ
R
are derived. We apply these results for the construction of quantum fields and for the study of scattering on the half line.
TL;DR: In this paper, the authors show how to use D and NS fivebranes in Type IIB superstring theory to construct large classes of finite N=1 supersymmetric four dimensional field theories.
Abstract: We show how to use D and NS fivebranes in Type IIB superstring theory to construct large classes of finite N=1 supersymmetric four dimensional field theories. In this construction, the beta functions of the theories are directly related to the bending of branes; in finite theories the branes are not bent, and vice versa. Many of these theories have multiple dimensionless couplings. A group of duality transformations acts on the space of dimensionless couplings; for a large subclass of models, this group always includes an overall $SL(2,\ZZ)$ invariance. In addition, we find even larger classes of theories which, although not finite, also have one or more marginal operators.
TL;DR: In this article, the authors use subfactor theory and algebraic quantum field theory to approach coset Conformal Field Theories (RCFTs) and prove a long-standing conjecture about the representations of these algebras.
Abstract: All unitary Rational Conformal Field Theories (RCFT) are conjectured to be related to unitary coset Conformal Field Theories, i.e., gauged Wess-Zumino-Witten (WZW) models with compact gauge groups. In this paper we use subfactor theory and ideas of algebraic quantum field theory to approach coset Conformal Field Theories. Two conjectures are formulated and their consequences are discussed. Some results are presented which prove the conjectures in special cases. In particular, one of the results states that a class of representations of coset $W_N$ ($N\geq 3$) algebras with critical parameters are irreducible, and under the natural compositions (Connes' fusion), they generate a finite dimensional fusion ring whose structure constants are completely determined, thus proving a long-standing conjecture about the representations of these algebras.
[...]
TL;DR: In this paper, the authors determined the double-pole divergence of O(p 6 ) in the meson sector of chiral perturbation theory, and then used an extension of this result to determine the p 6 contributions containing double chiral logarithm (L 2 ), single chiral (S 2 ) and products of two p 4 constants (L i r × L j r ) for F π, F K / Fπ, K l 3 and K e 4 form factors.
TL;DR: In this paper, the authors determined all real quadratic number fields with 2-class field tower of length at most 1, and all such fields were shown to have 1.5-approximation.
TL;DR: This work constructs infinite class field towers of global function fields with asymptotically many rational places and shows an improvement on the Gilbert‐Varshamov bound for linear codes over finite fields of a sufficiently large composite nonsquare order.
Abstract: We construct infinite class field towers of global function fields with asymptotically many rational places. In this way, we improve on asymptotic bounds of Serre, Perret, Schoof, and Xing. The results can be interpreted equivalently as asymptotic bounds on the number of rational points of smooth algebraic curves over finite fields. As an application, we show an improvement on the Gilbert-Varshamov bound for linear codes over finite fields of a sufficiently large composite nonsquare order.
TL;DR: The algebraic structure of thermo field dynamics lies in the q -deformation of the algebra of creation and annihilation operators as mentioned in this paper, and doubling of the degrees of freedom, tilde-conjugation rules, and Bogoliubov transformation for bosons and fermions are recognized as algebraic properties of h q (1) and h q(1|1) respectively.
TL;DR: In this paper, it was shown that R is necessarily the completion of a free algebra over an operad defined over a field of characteristic zero, where R is a cogroup in the category of complete algebras.
Abstract: Let \(\cal P\) be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete \({\cal P}\)-algebras. In this article, we show that R is necessarily the completion of a free \({\cal P}\)-algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad.
TL;DR: It is shown that the cyclic Eulerian elements linearly span a commutative semisimple algebra of QSn, which is naturally isomorphic to the algebra of the classical Euleria elements.
Abstract: LetSnbe the symmetric group on {1,?,n} and QSn its group algebra over the rational field; we assumen?2. ??Snis said a descent ini, 1?i?n-1, if ?(i)? (i+1); moreover, ? is said to have a cyclic descent if ?(n)?(1). We define the cyclic Eulerian elements as the sums of all elements inSnhaving a fixed global number of descents, possibly including the cyclic one. We show that the cyclic Eulerian elements linearly span a commutative semisimple algebra of QSn, which is naturally isomorphic to the algebra of the classical Eulerian elements. Moreover, we give a complete set of orthogonal idempotents for such algebra, which are strictly related to the usual Eulerian idempotents.
TL;DR: In this paper, the Hilbert function H(A/(h),-) in terms of A is bounded for generic linear forms of homogeneous K-algebras, where h ∈ A is a generic form.
Abstract: Let A be a homogeneous K-algebra where K is a field of characteristic 0, and h ∈ A a generic form. We bound the Hilbert function H(A/(h),-) in terms of H(A,-) which extends the bound given by M.Green for generic linear forms. We apply this to some conjectures from Higher Castelnuovo Theory and Cayley-Bacharach Theory.
TL;DR: In this article, a conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula and has applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions.
Abstract: On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik-Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories.