The Newton polytope of the discriminant of a quaternary cubic form

The Newton Polytope of the Morse Discriminant of a Univariate Polynomial

In this paper we compute the Newton polytope $\mathcal M_A$ of the Morse discriminant in the space of univariate polynomials with the given support set $A.$ Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of $\mathcal M_A.$

A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes

We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than $\lceil \frac {\dim P} 2 \rceil$.

Basecondary polytopes（基次级多胞体）

Basecondary polytopes

本文引入basecondary polytopes类多胞体，可统一研究次级多胞体、基多胞体及牛顿多胞体等，还涵盖无法归约为传统A-判别式与次级多胞体的Lyashko–Looijenga映射判别式，为相关几何研究提供统一框架。

Trianguloids and Triangulations of Root Polytopes

Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of r…

Newton Polytopes in Algebraic Combinatorics

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. Fo…