Equations for secant varieties of Chow varieties

TL;DR

This paper develops new equations for secant varieties of Chow varieties using prolongation, which could impact complexity theory and Valiant's conjecture by providing tools to measure polynomial complexity.

Contribution

It introduces a method to derive equations for secant varieties of Chow varieties as $GL(V)$-modules, advancing algebraic geometry tools relevant to computational complexity.

Findings

Derived equations for secant varieties of Chow varieties

Applied prolongation method to obtain $GL(V)$-module equations

Potential implications for Valiant's conjecture and complexity theory

Abstract

The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Finding equations in the ideal of secant varieties of Chow varieties would enable one to measure the complexity the permanent to prove Valiant's conjecture $\mathbf{VP\neq VNP}$. In this article, I use the method of prolongation to obtain equations for secant varieties of Chow varieties as $GL(V)$-modules.