Related papers: Box polynomials and the excedance matrix
A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…
The object of this paper is to give a systematic treatment of excedance-type polynomials. We first give a sufficient condition for a sequence of polynomials to have alternatingly increasing property, and then we present a systematic study…
The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…
Polynomials commute under composition are referred to as commuting polynomials. In this paper, we study division properties for commuting polynomials with rational (and integer) coefficients. As a consequence, we show an algebraic…
A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle…
Using PET and quantitative concatenation techniques, we establish box-norm control with the "expected" directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parameters.…
We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete…
We reconsider in modern terms the old discovery by A. Kirillov and M. Noumi, who devised peculiar operators adding columns to Young diagrams enumerating the Schur, Jack and Macdonald polynomials. In this sense, these are a kind of…
Just as knowing some roots of a polynomial allows one to factor it, a well-known result provides a factorization of any scalar differential operator given a set of linearly independent functions in its kernel. This note provides a…
We compute explicit formulas for the curvature operators and Poincar\'e polynomials of all compact irreducible symmetric spaces. We can easily derive the Poincar\'e polynomials using quantum numbers, giving a formula that mirrors the known…
A major factor contributing to the success of modern representation learning is the ease of performing various vector operations. Recently, objects with geometric structures (eg. distributions, complex or hyperbolic vectors, or regions such…
For operators generated by a certain class of infinite band matrices with matrix elements we establish a characterization of the resolvent set in terms of polynomial solutions of the underlying higher order finite difference equations. This…
A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics…
We consider partitions of n-dimensional boxes in R^n, n>1, into a finite number of boxes with pairwise disjoint interiors. We study sets X \subseteq (0,\infty) with the Property (W_n): for every n-dimensional box P and every partition of P,…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via…
This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…
We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined by directional derivatives, and prove it is…
Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate…
We find an explicit formula for the gamma vector in terms of the input polynomial in a way that extends it to arbitrary polynomials. More specifically, we find explicit linear combination in terms of coefficients of the input polynomial…