Equations for Chow Varieties, Their Secant Varieties and Other Varieties Arising in Complexity Theory

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

Secant varieties of toric varieties

Let $X_P$ be a smooth projective toric variety of dimension $n$ embedded in \PP^r using all of the lattice points of the polytope $P$. We compute the dimension and degree of the secant variety \Sec X_P. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties $X_A$ embedded using a set of lattice points A \subset P\cap\ZZ^n containing the vertices of $P$ and their nearest neighbors.Comment: v1, AMS LaTex, 5 figures, 25 pages; v2, reference added; v3, This is a major rewrite. We have strengthened our main results to include a classification of smooth lattice polytopes P such that Sec X_P does not have the expected dimension. (See Theorems 1.4 and 1.5.) There was also a considerable amount of reorganization, and some expository material was eliminated; v4, 28 pages, minor corrections, additional and updated reference