Flattenings and Koszul Young flattenings arising in complexity theory

TL;DR

This paper introduces new equations derived from flattenings and Koszul Young flattenings to improve lower bounds in algebraic complexity, specifically for symmetric border rank and depth 5 circuit size.

Contribution

It develops novel equations for Chow varieties and related varieties, leading to improved lower bounds in algebraic complexity theory.

Findings

New equations for Chow and secant varieties

Lower bounds for symmetric border rank of monomials

Lower bounds on depth 5 circuit size for the permanent

Abstract

I find new equations for Chow varieties, their secant varieties, and an additional variety that arises in the study of depth 5 circuits by flattenings and Koszul Young flattenings. This enables a new lower bound for symmetric border rank of $x_1x_2\cdots x_d$ when $d$ is odd, and a lower bound on the size of depth 5 circuits that compute the permanent.