Generating sequences and key polynomials

双图与二维函数域上非除数估值的生成序列

Dual Graphs and Generating Sequences of Non-divisorial Valuations on Two-dimensional Function Fields

本文基于Spivakovsky的双图理论，证明了二维函数域上非除数估值的最小生成序列，且定义方式与原用法不同，通过双图数据明确生成序列值。

生成秩1赋值的序列

Generating sequences of rank 1 valuations

本文研究三维正则局部环上的秩1赋值，提出构造其生成序列的算法，推广了此前仅适用于有理秩1赋值的算法，并给出有理秩3赋值的示例完成显式计算。

Strictness of the log-concavity of generating polynomials of matroids

Recently, it was proved by Anari-Oveis Gharan-Vinzant, Anari-Liu-Oveis Gharan-Vinzant and Brändén-Huh that, for any matroid $M$, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.

Toroidalization of generating sequences in dimension two function fields of positive characteristic

We give a characteristic free proof of the main result of our previous paper (math.AC/0509697) concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two dimensional algebraic regular local rings $R\subset S$ satisfies the conclusions of the Strong Monomialization theorem of Cutkosky and Piltant, the map between generating sequences in $R$ and $S$ has a toroidal structure.

多项式理想的低次极小生成集

Low-degree minimal generating sets of polynomial ideals

本文研究多项式环中有界次数生成元的理想，给出生成元最小数量的紧界和极小生成集最大规模的精确量化，并在多种域上构造达到紧界的例子，完整解决单变量情形并猜想多变量有限域渐近。

Symmetric edge polytopes and matching generating polynomials

Symmetric edge polytopes $\mathcal{A}_G$ of type A are lattice polytopes arising from the root system $A_n$ and finite simple graphs $G$. There is a connection between $\mathcal{A}_G$ and the Kuramoto synchronization model in physics. In particular, the normalized volume of $\mathcal {A}_G$ plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph $G$, we give a formula for the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ by using matching generating polynomials, where $\widehat{G}$ is the suspension of $G$. This gives also a formula…