Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility

Enumerative Aspects of Nullstellensatz Certificates

Using polynomial equations to model combinatorial problems has been a popular tool both in computational combinatorics as well as an approach to proving new theorems. In this paper, we look at several combinatorics problems modeled by systems of polynomial equations satisfying special properties. If the equations are infeasible, Hilbert's Nullstellensatz gives a certificate of this fact. These certificates have been studied and exhibit combinatorial meaning. In this paper, we generalize some known results and show that the Nullstellensatz certificate can be viewed as enumerating combinatorial …

Polynomials that vanish to high order on most of the hypercube

Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed $k\geq 1$ and $n$ large with respect to $k$, what is the minimum possible degree of a polynomial $P\in \mathbb{R}[x_1,\dots,x_n]$ with $P(0,\dots,0)\neq 0$ such that $P$ has zeroes of multiplicity at least $k$ at all points in $\{0,1\}^n\setminus \{(0,\dots,0)\}$? For $k=1$, a classical theorem of Alon and Füredi states that the minimum possible degree of such a polynomial equals $n$. In this paper, we solve the problem for all $k\ge…

无需烦恼的快速矩阵乘法：一种约束规划方法

Fast Matrix Multiplication Without Tears: A Constraint Programming Approach

本文提出约束规划方法求解快速矩阵乘法的约束满足问题，通过对称性破除、有效不等式、稀疏分解及求解器性能波动，发现算法或证明不可行，并在 CP Optimizer 上快速找到至 $3\ imes3$ 矩阵的算法。