Characterization of polynomials whose large powers have fully positive coefficients

Combinatorial Cellular Decompositions for the Space of Complex Coefficient Polynomials

We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we work out explicitly this classification for degree 3 polynomials, and other special families of polynomials. This work extends to the singular case similar considerations of Martin, Savitt, and Singer for non-singular basketballs.

拉伸舒伯特结构常数和关键系数的多项式性

Polynomiality of Stretched Schubert Structure Constants and Key Coefficients

研究关键多项式和舒伯特多项式仿射族中单项式系数的多项式性，通过结合德马祖尔算子等方法证明其最终为多项式，并扩展到有限乘积，利用舒伯特对偶性证明拉伸舒伯特结构常数最终是多项式，解决相关猜想。

Littlewood-Richardson coefficients for Grothendieck polynomials from integrability

We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant $K$-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-T…

Factorization of polynomials with analytic coefficients

We study monic univariate polynomials whose coefficients are analytic functions of a real variable and whose roots lie in a specified analytic curve. These include characteristic polynomials of unitary and hermitian matrices whose entries are analytic functions. We use a result of Newton to prove that every polynomial in such a class is a product of degree one polynomials in the class.

The leading coefficient of Lascoux polynomials

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 $δ(m,n,n-s)$ of the algebraic degree of semidefinite programming, and when $s=1$ we find its leading coefficient for types C, A and D.

舒伯特系数消失问题的概率多项式时间判定

Vanishing of Schubert coefficients in probabilistic polynomial time

本文从算法角度完整解决了舒伯特消失问题，证明舒伯特结构常数是否为零可在概率多项式时间内判定。