Root statistics of random polynomials with bounded Mahler measure

The Mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coefficient times the modulus of the product of its roots outside the unit circle. The roots of a real degree N polynomial chosen uniformly from the set of polynomials of Mahler measure at most 1 yield a Pfaffian point process on the complex plane. When N is large, with probability tending to 1, the roots tend to the unit circle, and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle. When this point is away from the real axis (on which there is a positive probability of finding a root) the scaled process degenerates to a determinantal point process with the same local statistics (i.e.   scalar kernel) as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of Mahler measure at most 1. Three new matrix kernels appear in a neighborhood of ±1 which encode information about the correlations between real roots, between complex roots and between real and complex roots. Away from the unit circle, the kernels converge to new limiting kernels, which imply among other things that the expected number of roots in any open subset of C$\mathbb{C}$ disjoint from the unit circle converges to a positive number. We also give ensembles with identical statistics drawn from two-dimensional electrostatics with potential theoretic weights, and normal matrices chosen with regard to their topological entropy as actions on Euclidean space.     

Sensitivity of convergent matrices and polynomials

A matrix A is said to be convergent if and only if all its characteristic roots have modulus less than unity. When A is real an explicit expression is given for real matrices B such that A + B is also convergent, this expression depending upon the solution of a quadratic matrix equation of Riccati type. If A and A + B are taken to be in companion form, then the result becomes one of convergent polynomials (i.e., polynomials whose roots have modulus less then unity), and is much easier to apply. A generalization is given for the case when A and A + B are complex and have the same number of roots inside and outside a general circle.     

关于Sturm定理的一点注记

经典的S turm定理用于判定多项式在给定区间上不同的实根个数,但是并不能刻画重根的情况.在这里定义了推广的S turm序列,将S turm定理进行一定地延拓,给出区间上多项式的所有实根均是偶重根或奇重根的充要条件.作为应用,讨论了多项式正(负)半定的判定问题.     

Asymptotic constants for the error of Hermite-Fejér interpolation on the unit circle

The paper deals with the rate of convergence for the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system the n roots of a complex number with modulus one. The order of convergence and the asymptotic constants are obtained when we consider analytic functions on open disks and open annulus containing the unit circle.     

Majorization and multiplier sequences

We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link between the applications.     

Polynomials with all their roots on <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>

We give a characterization of monic polynomials with coefficients in the ring of integers of a Galois number field having all of their roots on the unit circle. Such a characterization is given in terms of finitely many sums of powers of the roots of the considered polynomials.     

On the number of distinct roots of a lacunary polynomial over finite fields

We obtain new upper bounds on the number of distinct roots of lacunary polynomials over finite fields. Our focus will be on polynomials for which there is a large gap between consecutive exponents in the monomial expansion.     

Equimodular curves

This paper is motivated by a problem that arises in the study of partition functions of Potts models, including as a special case chromatic polynomials. When the underlying graphs have the form of ‘bracelets’, the chromatic polynomials can be expressed in terms of the eigenvalues of a matrix. In this situation a theorem of Beraha, Kahane and Weiss asserts that the zeros of the polynomials approach the curves on which the matrix has two eigenvalues with equal modulus. It is shown here that (in general) these ‘equimodular’ curves comprise a number of segments, the end-points of which are the roots (possibly coincident) of a polynomial equation. The equation represents the vanishing of a discriminant, and the segments are in bijective correspondence with the double roots of another polynomial equation, which is significantly simpler than the discriminant equation. Singularities of the segments can occur, corresponding to the vanishing of a Jacobian. In addition, it is proved by algebraic means that the equimodular curves for a reducible matrix are closed curves. The question of dominance is investigated, and a method of constructing the dominant equimodular curves for a reducible matrix is suggested. These results are illustrated by explicit calculations in a specific case.     

Tchebyshev triangulations of stable simplicial complexes

We generalize the notion of the Tchebyshev transform of a graded poset to a triangulation of an arbitrary simplicial complex in such a way that, at the level of the associated F-polynomials j_∑fj−1(j(x−1)/2), the triangulation induces taking the Tchebyshev transform of the first kind. We also present a related multiset of simplicial complexes whose association induces taking the Tchebyshev transform of the second kind. Using the reverse implication of a theorem by Schelin we observe that the Tchebyshev transforms of Schur stable polynomials with real coefficients have interlaced real roots in the interval (−1,1), and present ways to construct simplicial complexes with Schur stable F-polynomials. We show that the order complex of a Boolean algebra is Schur stable. Using and expanding the recently discovered relation between the derivative polynomials for tangent and secant and the Tchebyshev polynomials we prove that the roots of the corresponding pairs of derivative polynomials are all pure imaginary, of modulus at most one, and interlaced.     

On One Method for Factorization of Algebraic Polynomials

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.     

Notes on the Roots of Ehrhart Polynomials

We determine lattice polytopes of smallest volume with a given number of interior lattice points. We show that the Ehrhart polynomials of those with one interior lattice point have largest roots with norm of order n², where n is the dimension. This improves on the previously best known bound n and complements a recent result of Braun where it is shown that the norm of a root of a Ehrhart polynomial is at most of order n². For the class of 0-symmetric lattice polytopes we present a conjecture on the smallest volume for a given number of interior lattice points and prove the conjecture for crosspolytopes. We further give a characterisation of the roots of Ehrhart polyomials in the three-dimensional case and we classify for n ≤ 4 all lattice polytopes whose roots of their Ehrhart polynomials have all real part -1/2. These polytopes belong to the class of reflexive polytopes.     

Gale duality bounds for roots of polynomials with nonnegative coefficients

We bound the location of roots of polynomials that have nonnegative coefficients with respect to a fixed but arbitrary basis of the vector space of polynomials of degree at most d. For this, we interpret the basis polynomials as vector fields in the real plane, and at each point in the plane analyze the combinatorics of the Gale dual vector configuration. This approach permits us to incorporate arbitrary linear equations and inequalities among the coefficients in a unified manner to obtain more precise bounds on the location of roots. We apply our technique to bound the location of roots of Ehrhart and chromatic polynomials. Finally, we give an explanation for the clustering seen in plots of roots of random polynomials.     

Kronecker's theorem and Lehmer's problem for polynomials in several variables

Mahler defined the measure of a polynomial in several variables to be the geometric mean of the modulus of the polynomial averaged over the torus. The classical theorem of Kronecker which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk can be regarded as characterizing those polynomials of one variable whose measure is exactly 1. Here this result is generalized to polynomials in several variables. The method employed also gives easy generalizations of recent results of Schinzel and Dobrowolski on Lehmer's problem.     

A problem of Boyd concerning geometric means of polynomials

In this paper we derive an equality which characterizes the distribution of the modulus of a polynomial on the unit circle. This inequality is used to prove a conjecture of Boyd concerning the geometric mean of the modulus of a polynomial of several variables averaged over the torus. References are cited which discuss the relationship of this conjecture to a classical question of Lehmer concerning the distribution of roots of polynomials.     

On Descartes' rules of signs and their exactness

This paper revisits the Descartes' rules of signs and provides new bounds for the number of complex roots of a polynomial in certain complex regions. We also prove that the Descartes' rules associated with the Bernstein basis are exact for polynomials whose roots are real. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Degree of Approximation for Bivariate Chlodowsky–Szasz–Charlier Type Operators

We consider a combination of Chlodowsky polynomials with generalized Szasz operators involving Charlier polynomials. We give the degree of approximation for these bivariate operators by means of the complete and partial modulus of continuity, and also by using weighted modulus of continuity. Furthermore, we construct a GBS (Generalized Boolean Sum) operator of bivariate Chlodowsky–Szasz–Charlier type and estimate the order of approximation in terms of mixed modulus of continuity.     

一类新的渐近稳定多项式

[1]对[2]中提出的求多项式根的渐近因子分离法进行了详细的讨论,国外对此法(称为Sebastiao e Silva算法)也进行了大量的研究,例如[3]—[9].此算法有不少令人注目的特点.本文将讨论一类新的渐近稳定多项式,它们也具有许多与 Sebastiao e Silva算法相类似的性质.     

Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.     

Properties of a generalized Sylvester matrix

We consider problems close to that of the minimal stabilization of a linear vector (i.e., MISO or SIMO) dynamic system; more specifically, the problem of determining the number of common roots of a family of polynomials, and investigating the properties of the so-called generalized Sylvester matrix. The classical definition of the Sylvester matrix is valid for two polynomials, and there are different methods for defining the generalized (extended) Sylvester matrix for a family of polynomials. In this work, we consider a definition of the generalized Sylvester matrix and its properties in the context of their potential future application for solving the minimal stabilization problem.     

Exceptional units in a family of quartic number fields

We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a one-parameter family of polynomials with two real roots.