Entropy of radical ideal of a tropical prevariety

Complexity of deciding whether a tropical linear prevariety is a tropical variety

We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization $A^\perp$ and the double tropical orthogonalization $A^{\perp \perp}$ of a subset $A$ of the vector space $({\mathbb R} \cup \{ \infty \})^n$. We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.

1维热带预簇的度数

Degree of 1-dimensional tropical prevariety

本文定义了热带预簇的热带 Hilbert 函数与热带度，证明了1维情况下该度数极限的存在性，并显式计算了星形预簇的热带度变体。

热带希尔伯特多项式的一个版本（一维情形）

A tropical version of Hilbert polynomial (in dimension one)

本文为一维热带多项式定义热带希尔伯特函数，证明其大次数时等于线性函数与整数周期函数之和，并给出热带熵的精确界。

Computing Tropical Prevarieties in Parallel

The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver. We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved…

Dimension computations for tropical determinantal varieties and prevarieties

This paper proves that when the $r \times r$ minors of an $m \times n$ matrix of indeterminates are not a tropical basis then the tropical prevariety has greater dimension than the tropical variety. It proves the same for the $r \times r$ minors of an $n \times n$ symmetric matrix of indeterminates when r > 4.

多元超度量根计数

Multivariate ultrametric root counting

设$K$是一个关于离散非阿基米德赋值完备的域，$k$是剩余域。考虑$K\vars$中$n$多项式方程的系统$F$。我们的第一个结果是一个重新制定的经典亨泽尔引理的语言热带几何：我们证明了充分条件（半正则性$w$），保证第一位数映射$δ:(K^\ast)^n\to(k^\ast)^n$是一个一对一的对应关系的解决方案$F$在$(K^\ast)^n$与估值$w$和解决方案在$(k^\ast)^n$的初始形式系统${\rm in}_w(F)$。利用这一结果，我们提供了一个明确的公式的解决方案在$(K^\ast)^n$的某一类系统的多项式方程（称为定期），其特征在于有有限的热带prevariety，由有初始形式仅由二项式，并在热带prevariety中的任何一点是半正则的。最后，作为根计数公式的结果，我们得到了期望的根数在$(K^\ast)$的一元多项式与给定的支持和随机系数。