A Polynomial Degree Bound on Equations of Non-rigid Matrices and Small Linear Circuits

Abstract

We show that there is a defining 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 n 2 -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 sparsity at most n 2 /100 satisfies P (M ) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [GHIL16] and improves the best upper bound known for this problem down from exp(n 2 ) [KLPS14, GHIL16] 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 n 2 /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 [SV15] 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.

If VNP is hard, then so are equations for it

“…In the context of equations for varieties in algebraic complexity, Kumar and Volk [KV20] proved polynomial degree bounds on the equations of the Zariski closure of the set of non-rigid matrices as well as small linear circuits over all large enough fields.…”