A Polynomial Degree Bound on Equations for Non-Rigid Matrices and Small Linear Circuits

Abstract

We show that there is an equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field 𝔽, there is a non-zero n²-variate polynomial P ∈ 𝔽[x_{1, 1}, …, x_{n, n}] of degree at most poly(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and a matrix of sparsity at most n²/100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [Fulvio Gesmundo et al., 2016] and improves the best upper bound known for this problem down from exp(n²) [Abhinav Kumar et al., 2014; Fulvio Gesmundo et al., 2016] to poly(n). 
We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n²/200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. 
Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [Amir Shpilka and Ilya Volkovich, 2015] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof. 
As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [Valentine Kabanets and Russell Impagliazzo, 2004].