The leading coefficient of Lascoux polynomials

TL;DR

This paper determines the leading coefficients and degrees of Lascoux polynomials across types C, A, and D, with applications to algebraic degrees in semidefinite programming.

Contribution

It explicitly computes the leading coefficients and degrees of Lascoux polynomials for various matrix types, extending their applications.

Findings

Leading coefficient of type C Lascoux polynomials identified.

Degrees of Lascoux polynomials determined for types C, A, D.

Degree of algebraic degree polynomial in semidefinite programming found.

Abstract

Lascoux polynomials have been recently introduced to prove polynomiality of the maximum-likelihood degree of linear concentration models. We find the leading coefficient of the Lascoux polynomials (type C) and their generalizations to the case of general matrices (type A) and skew symmetric matrices (type D). In particular, we determine the degrees of such polynomials. As an application, we find the degree of the polynomial $\delta(m,n,n-s)$ of the algebraic degree of semidefinite programming, and when $s=1$ we find its leading coefficient for types C, A and D.