Related papers: Boolean product polynomials and Schur-positivity

This paper considers some work done by the author and Catlin [CD1,CD2,CD3] concerning positivity conditions for bihomogeneous polynomials and metrics on bundles over certain complex manifolds. It presents a simpler proof of a special case…

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…

The central binomial series at negative integers are expressed as a linear combination of values of certain two polynomials. We show that one of the polynomials is a special value of the bivariate Eulerian polynomial and the other…

We develop algebro-combinatorial tools for computing the Thom polynomials for the Morin singularities $A_i(-)$ ($i\ge 0$). The main tool is the function $F^{(i)}_r$ defined as a combination of Schur functions with certain numerical…

We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the…

We give the Thom polynomials for the singularities $I_{2,2}$ associated with maps $({\bf C}^{\bullet},0) \to ({\bf C}^{\bullet+k},0)$ with parameter $k\ge 0$. Our computations combine the characterization of Thom polynomials via the…

We study correlation functions of the characteristic polynomials in coupled matrix models based on the Schur polynomial expansion, which manifests their determinantal structure.

We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…

A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of…

Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we…

We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of…

We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…

We study a new kind of symmetric polynomials P_n(x_1,...,x_m) of degree n in m real variables, which have arisen in the theory of numerical semigroups. We establish their basic properties and find their representation through the power sums…

We introduce partial differential encodings of Boolean functions as a way of measuring the complexity of Boolean functions. These encodings enable us to derive from group actions non-trivial bounds on the Chow-Rank of polynomials used to…

Given an element in a finite-dimensional real vector space, $V$, that is a nonnegative linear combination of basis vectors for some basis $B$, we compute the probability that it is furthermore a nonnegative linear combination of basis…

A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability…

We study decreasing binary trees in which every vertex with two children is colored red or blue. We construct two bijections. The first, to ordered set partitions into odd-sized blocks each arranged as an alternating permutation, shows that…

Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur-positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets…

We investigate asymptotic polynomial approximation for a class of weighted Bloch functions in the unit disc. Our main result is a structural theorem on asymptotic polynomial approximation in the unit disc, in the flavor of the classical…