Convergence in measure of multivariate Pade approximants

Convergence conclusions of Padé approximants in the univariate case can be found in various papers. However, results in the multivariate case are few. A. Cuyt seems to be the only one who discusses convergence for multivariate Pade approximants, she gives in [2] a de Montessus de Bollore type theorem. In this paper, we will discuss the zero set of a real multivariate polynomial, and present a convergence theorem in measure of multivariate Padé approximant. The proof technique used in this paper is quite different from that used in the univariate case. Supported by National Science Foundation of China for Youth     

Mixed multiplicities of ideals versus mixed volumes of polytopes

The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's theorem which asserts that the number of common zeros of a system of Laurent polynomial equations in the torus is bounded above by the mixed volume of their Newton polytopes.

     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].     

A new approach to constant term identities and Selberg-type integrals

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. Built on a recent idea of Karasev and Petrov we develop a general interpolation based method that is powerful enough to establish many such identities in a simple manner. The main consequence is the proof of a conjecture of Forrester related to the Calogero–Sutherland model. In fact we prove a more general theorem, which includes Aomoto's constant term identity at the same time. We also demonstrate the relevance of the method in additive combinatorics.     

Laurent polynomials and Eulerian numbers

Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels poses two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweil?s theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

A higher-dimensional Kurzweil theorem for formal Laurent series over finite fields

In a recent paper, Kim and Nakada proved an analogue of Kurzweilʼs theorem for inhomogeneous Diophantine approximation of formal Laurent series over finite fields. Their proof used continued fraction theory and thus cannot be easily extended to simultaneous Diophantine approximation. In this note, we give another proof which works for simultaneous Diophantine approximation as well.     

Generalized matrix functions and determinants

In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.     

An extended version of Schur-Cohn-Fujiwara theorem in stability theory

This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.     

Remarks on Hilbert series of graded modules over polynomial rings

In this article we discuss a result on formal Laurent series and some of its implications for Hilbert series of finitely generated graded modules over standard-graded polynomial rings: For any integer Laurent function of polynomial type with non-negative values the associated formal Laurent series can be written as a sum of rational functions of the form ${\frac{Q_j(t)}{(1-t)^j}}$ , where the numerators are Laurent polynomials with non–negative integer coefficients. Hence any such series is the Hilbert series of some finitely generated graded module over a suitable polynomial ring ${\mathbb{F}[X_1 , \ldots , X_n]}$ . We give two further applications, namely an investigation of the maximal depth of a module with a given Hilbert series and a characterization of Laurent polynomials which may occur as numerator in the presentation of a Hilbert series as a rational function with a power of (1 ? t) as denominator.     

The Abhyankar-Jung theorem revisited

In this paper we use a technique introduced by P.M. Cohn in order to prove the so-called Abhyankar–Jung theorem, i.e. the existence of parametric equations for a quasiordinary hypersurface. The basic tool of the proof is the equivalence between the existence of solutions for a polynomial equation and the existence of an eigenvalue of its companion matrix.     

Partial Skew Polynomial Rings and Jacobson Rings

In this article we consider rings R with a partial action α of an infinite cyclic group G on R. We generalize the well-known results about Jacobson rings and strongly Jacobson rings in skew polynomial rings and skew Laurent polynomial rings to partial skew polynomial rings and partial skew Laurent polynomial rings.     

Perfect matchings and the octahedron recurrence

We study a recurrence defined on a three dimensional lattice and prove that its values are Laurent polynomials in the initial conditions with all coefficients equal to one. This recurrence was studied by Propp and by Fomin and Zelivinsky. Fomin and Zelivinsky were able to prove Laurentness and conjectured that the coefficients were 1. Our proof establishes a bijection between the terms of the Laurent polynomial and the perfect matchings of certain graphs, generalizing the theory of Aztec Diamonds. In particular, this shows that the coefficients of this polynomial, and polynomials obtained by specializing its variables, are positive, a conjecture of Fomin and Zelevinsky.     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

Newton polygons and curve gonalities

We give a combinatorial upper bound for the gonality of a curve that is defined by a bivariate Laurent polynomial with given Newton polygon. We conjecture that this bound is generically attained, and provide proofs in a considerable number of special cases. One proof technique uses recent work of M. Baker on linear systems on graphs, by means of which we reduce our conjecture to a purely combinatorial statement.     

Radicals in skew polynomial and skew Laurent polynomial rings

We provide a general procedure for characterizing radical-like functions of skew polynomial and skew Laurent polynomial rings under grading hypotheses. In particular, we are able to completely characterize the Wedderburn and Levitzki radicals of skew polynomial and skew Laurent polynomial rings in terms of ideals in the coefficient ring. We also introduce the T-nilpotent radideals, and perform similar characterizations.     

On the group extension of the transformation associated to non-archimedean continued fractions

Summary Let F_q be a finite field with q elements. We consider formal Laurent series of F_q -coefficients with their continued fraction expansions by F_q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.     

Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

A Toeplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Johansson.     

A probabilistic proof of the fundamental theorem of algebra

We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

     

The complexity of the equivalence and equation solvability problems over nilpotent rings and groups

It is proved that the equation solvability problem can be solved in polynomial time for finite nilpotent rings. Ramsey’s theorem is employed in the proof. Then, using the same technique, a theorem of Goldmann and Russell is reproved: the equation solvability problem can be solved in polynomial time for finite nilpotent groups.