Equations for Chow Varieties, Their Secant Varieties and Other Varieties Arising in Complexity Theory
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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.
Koszul Young flattening
VP and VNP
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
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