Showing papers on "Field (mathematics) published in 2000"

Determinantal random point fields

Abstract: This paper contains an exposition of both recent and rather old results on determinantal random point fields. We begin with some general theorems including proofs of necessary and sufficient conditions for the existence of a determinantal random point field with Hermitian kernel and of a criterion for weak convergence of its distribution. In the second section we proceed with examples of determinantal random fields in quantum mechanics, statistical mechanics, random matrix theory, probability theory, representation theory, and ergodic theory. In connection with the theory of renewal processes, we characterize all Hermitian determinantal random point fields on and with independent identically distributed spacings. In the third section we study translation-invariant determinantal random point fields and prove the mixing property for arbitrary multiplicity and the absolute continuity of the spectra. In the last section we discuss proofs of the central limit theorem for the number of particles in a growing box and of the functional central limit theorem for the empirical distribution function of spacings.

Determinants of block matrices

Abstract: Let us first consider the 2 x 2 matrices and Their sum and product are given by Here the entries a, b, c, d, e, f, g, h can come from a field, such as the real numbers, or more generally from a ring, commutative or not.

Polynomials with Special Regard to Reducibility

Abstract: This book covers most of the known results on reducibility of polynomials over arbitrary fields, algebraically closed fields and finitely generated fields. Results valid only over finite fields, local fields or the rational field are not covered here, but several theorems on reducibility of polynomials over number fields that are either totally real or complex multiplication fields are included. Some of these results are based on recent work of E. Bombieri and U. Zannier (presented here by Zannier in an appendix). The book also treats other subjects like Ritt's theory of composition of polynomials, and properties of the Mahler measure, and it concludes with a bibliography of over 300 items. This unique work will be a necessary resource for all number theorists and researchers in related fields.

Singularities of plane curves

Abstract: This book provides a comprehensive and self-contained exposition of the algebro-geometric theory of singularities of plane curves, covering both its classical and its modern aspects. The book gives a unified treatment, with complete proofs, presenting modern results which have only ever appeared in research papers. It updates and correctly proves a number of important classical results for which there was formerly no suitable reference, and includes new, previously unpublished results as well as applications to algebra and algebraic geometry. This book will be useful as a reference text for researchers in the field. It is also suitable as a textbook for postgraduate courses on singularities, or as a supplementary text for courses on algebraic geometry (algebraic curves) or commutative algebra (valuations, complete ideals).

Symplectic Fermions

Abstract: We study two-dimensional conformal field theories generated from a ``symplectic fermion'' - a free two-component fermion field of spin one - and construct the maximal local supersymmetric conformal field theory generated from it This theory has central charge c=-2 and provides the simplest example of a theory with logarithmic operators Twisted states with respect to the global SL(2,C)-symmetry of the symplectic fermions are introduced and we describe in detail how one obtains a consistent set of twisted amplitudes We study orbifold models with respect to finite subgroups of SL(2,C) and obtain their modular invariant partition functions In the case of the cyclic orbifolds explicit expressions are given for all two-, three- and four-point functions of the fundamental fields The C_2-orbifold is shown to be isomorphic to the bosonic local logarithmic conformal field theory of the triplet algebra encountered previously We discuss the relation of the C_4-orbifold to critical dense polymers

Octonions, Jordan Algebras and Exceptional Groups

Abstract: The 1963 Gottingen notes of T. A. Springer are well-known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. In the group-theoretical part use is made of some results from the theory of linear algebraic groups. The book will be useful to mathematicians interested in octonion algebras and Albert algebras, or in exceptional groups. It is suitable for use in a graduate course in algebra.

On Subfields of the Hermitian Function Field

Abstract: The Hermitian function field H D K.x;y/ is defined by the equation y q CyD x qC1 (q being a power of the characteristic of K). Over K D F q2 it is a maximal function field; i.e. the numberN.H/ofF q 2 -rational places attains the Hasse-Weil upper boundN.H/Dq 2 C1C2g.H/q. All subfields K $ EH are also maximal. In this paper we construct a large number of nonrational subfields E H , by considering the fixed fields H G under certain groups G of automorphisms of H=K. Thus we obtain many integersg> 0 that occur as the genus of some maximal function field overF q2 . Mathematics Subject Classifications ( 1991): 11Gxx, 14Gxx

$D$-brane physics and noncommutative Yang–Mills theory

Abstract: We discuss the physics of a single Dp-brane in the presence of a background electromagnetic field B_{ij}. It has recently been shown \cite{SW} that, in a specific \alpha ' \to 0 limit, the physics of the brane is correctly described by noncommutative Yang-Mills theory, where the noncommutative gauge potential is given explicitly in terms of the ordinary U(1) field. In a previous paper \cite{SC} the physics of a D2-brane was analyzed in the Sen-Seiberg limit of M(atrix) theory by considering a specific coordinate change on the brane world-volume. We show in this note that the limit considered in \cite{SC} is the same as the one described in \cite{SW}, in the specific case p=2, rk B_{ij} = 2. Moreover we show that the coordinate change in \cite{SC} can be reinterpreted, in the spirit of \cite{SW}, as a field redefinition of the ordinary Yang-Mills field, and we prove that the transformations agree for large backgrounds. The results are finally used to considerably streamline the proof of the equivalence of the standard Born-Infeld action with noncommutative Yang-Mills theory, in the large wave-length regime.

Banach space representations and Iwasawa theory

Abstract: The lack of a $p$-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic field $K$. For example, Diarra showed that the abelian group $G=\dZ$ has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations. We therefore address the problem of finding an additional ''finiteness'' condition on such representations that will lead to a reasonable theory. We introduce such a condition that we call ''admissibility''. We show that the category of all admissible $G$-representations is reasonable -- in fact, it is abelian and of a purely algebraic nature -- by showing that it is anti-equivalent to the category of all finitely generated modules over a certain kind of completed group ring $K[[G]]$. As an application of our methods we determine the topological irreducibility as well as the intertwining maps for representations of $GL_2(\dZ)$ obtained by induction of a continuous character from the subgroup of lower triangular matrices.

Model Theory of Differential Fields

Abstract: This article surveys the model theory of differentially closed fields, an interesting setting where one can use model-theoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications. A differential field is a field K equipped with a derivation δ : K → K; recall that this means that, for x, y ∈ K, we have δ(x + y) = δ(x) + δ(y) and δ(xy) = x δ(y) + yδ(x). Roughly speaking, such a field is called differentially closed when it contains enough solutions of ordinary differential equations. This setting allows one to use model-theoretic methods, and particularly dimensiontheoretic ideas, to obtain interesting algebraic information. In this lecture I give a survey of the model theory of differentially closed fields, concluding with an example — Hrushovski’s proof of the Mordell–Lang conjecture in characteristic zero — showing how model-theoretic methods in this area can be used in diophantine applications. I will not give the proofs of the main theorems. Most of the material in Sections 1–3 can be found in [Marker et al. 1996], while the material in Section 4 can be found in [Hrushovski and Sokolovic ≥ 2001; Pillay 1996]. The primary reference on differential algebra is [Kolchin 1973], though the very readable [Kaplansky 1957] contains most of the basics needed here, as does the more recent [Magid 1994]. The book [Buium 1994] also contains an introduction to differential algebra and its connections to diophantine geometry. We refer the reader to these sources for references to the original literature. 1. Differentially Closed Fields Throughout this article all fields will have characteristic zero. A differential field is a field K equipped with a derivation δ : K → K. The field of constants is C = {x ∈ K : δ(x) = 0}. We will study differential fields using the language L = {+,− , · , δ, 0, 1}, the language of rings augmented by a unary function symbol δ. The theory of

Hyperelliptic jacobians without complex multiplication

Abstract: has only trivial endomorphisms over an algebraic closure of the ground field K if the Galois group Gal(f) of the polynomial f ∈ K[x] is “very big”. More precisely, if f is a polynomial of degree n ≥ 5 and Gal(f) is either the symmetric group Sn or the alternating group An then End(J(C)) = Z. Notice that it easily follows that the ring of K-endomorphisms of J(C) coincides with Z and the real problem is how to prove that every endomorphism of J(C) is defined over K. There are some results of this type in the literature. Previously Mori [8], [9] has constructed explicit examples (in all characteristics) of hyperelliptic jacobians without nontrivial endomorphisms. In particular, he provided examples over Q with semistable Cf and big (doubly transitive) Gal(f) [9]. The semistability of Cf implies the semistability of J(Cf ) and, thanks to a theorem of Ribet [14], all endomorphisms of J(Cf ) are defined over Q. (Applying to Cf/Q the Shafarevich conjecture [17] (proven by Fontaine [3] and independently by Abrashkin [1], [2]) and using Lemma 4.4.3 and arguments on p. 42 of [16], one may prove that the Galois group Gal(f) of the polynomial f involved is S2g+1 where deg(f) = 2g + 1.) Andre ([7], pp. 294-295) observed that results of Katz ([5], [6]) give rise to examples of hyperelliptic jacobians J(Cf ) over the field of rational function C(z) with End(J(Cf )) = Z. Namely, one may take f(x) = h(x)−z where h(x) ∈ C[x] is a Morse function. In particular, this explains Mori’s example [8]

A Survey of Diophantine Approximation in Fields of Power Series

Abstract: The fields of power series (or perhaps better called formal numbers) are analogues of the field of real numbers. Many questions in number theory which have been studied in the setting of the real numbers can be transposed to the setting of the power series. The study of rational approximation to algebraic real numbers has been intensively developped starting from the middle of the nineteenth century with the work of Liouville up to the celebrated theorem of Roth established in 1955. In the last thirty years, several mathematicians have studied diophantine approximation in fields of power series. We present here a summary of the present knowledge on this subject, emphasizing the analogies and differences with the situation in the real numbers case.

Closure of Periodic Points Over a Non-Archimedean Field

Abstract: The closure of the periodic points of rational maps over a non-archimedean field is studied. An analogue of Montel's theorem over non-archimedean fields is first proved. Then, it is shown that the (nonempty) Julia set of a rational map over a non-archimedean field is contained in the closure of the periodic points.

Finiteness properties of local cohomology modules for regular local rings of mixed characteristic: the unramified case

Abstract: In this note we partially extend results on the finite-ness properties of local cohomology modules from the case of a regular local ring containing a field to the unramified case of a regular local ring of mixed characteristic.

Arithmetic height functions over finitely generated fields

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 prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture).

Shifting Operations and Graded Betti Numbers

Abstract: The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal.

Matroids, motives and conjecture of Kontsevich

Abstract: Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchhoff polynomials to the representation spaces of matroids. Moreover, using Mnev's universality theorem, we show that these schemes essentially generate all arithmetic of schemes of finite type over the integers.

Using number fields to compute logarithms in finite fields

Abstract: We describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field. We conjecture that the running time of the algorithm, when restricted to finite fields of an arbitrary but fixed degree, is L q [1/3; (64/9) 1/3 + o(1)], where q is the cardinality of the field, L q [s; c] = exp(c(log q) s (log log q) 1- s ), and the o(1) is for q → ∞. The number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time.

Triangular Hopf algebras with the Chevalley property

Abstract: We say that a Hopf algebra has the Chevalley property if the tensor product of any two simple modules over this Hopf algebra is semisimple. In this paper we classify finite dimensional triangular Hopf algebras with the Chevalley property, over the field of complex numbers. Namely, we show that all of them are twists of triangular Hopf algebras with R-matrix having rank <=2, and explain that the latter ones are obtained from group algebras of finite supergroups by a simple modification procedure. We note that all examples of finite dimensional triangular Hopf algebras which are known to the authors, do have the Chevalley property, so one might expect that our classification potentially covers all finite dimensional triangular Hopf algebras.

Automorphisms, non-linear realizations and branes

Abstract: The theory of non-linear realizations is used to derive the dynamics of the branes of M-theory. A crucial step in this procedure is to use the enlarged automorphism group of the supersymmetry algebra recently introduced. The field strengths of the worldvolume gauge fields arise as some of the Goldstone fields associated with this automorphism group. The relationship to the superembedding approach is given.

On the Kleinian construction of Abelian functions of canonical algebraic curves

Abstract: The discovery of classical and quantum completely integrable systems led to an increase in interest in the theory of Abelian functions in theoretical physics and applied mathematics. This area was considered traditionally as a field of pure mathematics. This new trend makes it necessary to reconsider classical results in the area from the point of view of modern applications. In this paper we consider an arbitrary algebraic curve V of genus g and construct the field of meromorphic functions on its Jacobi variety Jac V in terms of Kleinian -functions,

A probabilistic algorithm to test local algebraic observability in polynomial time

Abstract: The following questions are often encountered in system and control theory. Given an algebraic model of a physical process, which variables can be, in theory, deduced from the input-output behavior of an experiment? How many of the remaining variables should we assume to be known in order to determine all the others? These questions are parts of the \emph{local algebraic observability} problem which is concerned with the existence of a non trivial Lie subalgebra of the symmetries of the model letting the inputs and the outputs invariant. We present a \emph{probabilistic seminumerical} algorithm that proposes a solution to this problem in \emph{polynomial time}. A bound for the necessary number of arithmetic operations on the rational field is presented. This bound is polynomial in the \emph{complexity of evaluation} of the model and in the number of variables. Furthermore, we show that the \emph{size} of the integers involved in the computations is polynomial in the number of variables and in the degree of the differential system. Last, we estimate the probability of success of our algorithm and we present some benchmarks from our Maple implementation.

Constructing Recursion Operators in Intuitionistic Type Theory

Abstract: Martin-L\"of's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. Included are < on natural numbers, and relations formed by inverse images, addition, multiplication, and exponentiation of other relations. The constructions are given in full detail to allow their use in theorem provers for Type Theory, such as Nuprl. The theory is compared with work in the field of ordinal recursion over higher types.

Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions

Abstract: Let ƒ1, … , ƒs be polynomials in n variables over a field of characteristic zero and d be the maximum of their total degree. We propose a new probabilistic algorithm for computing a geometric resolution of each equidimensional part of the variety defined by the system ƒ1 = ··· = ƒs = 0. The returned resolutions are encoded by means of Straight-Line Programs and the complexity of the algorithm is polynomial in a geometric degree of the system. In the worst case this complexity is asymptotically polynomial in sdn.

Group Averaging and Refined Algebraic Quantization: Where are we now?

Abstract: Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems. They give constructive algorithms for generating observables and the physical inner product. This work outlines the current status of these ideas with an eye toward quantum gravity. The main goal is provide a description of outstanding problems and possible research topics in the field.