Equations for Chow Varieties, Their Secant Varieties and Other Varieties Arising in Complexity Theory
Abstract
The Chow variety of polynomials that decompose as a product of linear forms has been studied for more than 100 years. Brill, Gordon, and others obtained set-theoretic equations for the Chow variety. I compute Brill's equations as a GL (V )-module. I find new equations for Chow varieties, their secant varieties, and an additional variety by flattenings and Koszul Young flattenings. This enables a new lower bound for the symmetric border rank of x1x2 ··· xd when d is odd and a new complexity lower bound for the permanent. I use the method of prolongation to obtain equations for secant varieties of Chow varieties as GL(V )-modules. The goal of studying these varieties arising in complexity theory is to separate VP from VNP, which is an algebraic analog of the famous P versus NP problem.
Subject
Chow varietyBrill's equations
secant variety
flattening
Koszul Young flattening
permanent
VP and VNP
Citation
Guan, Yonghui (2016). Equations for Chow Varieties, Their Secant Varieties and Other Varieties Arising in Complexity Theory. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /157875.
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