Jacobian Hits Circuits: Hitting Sets, Lower Bounds for Depth-$D$ Occur-$k$ Formulas and Depth-3 Transcendence Degree-$k$ Circuits

Abstract

We present a single common tool to strictly subsume all known cases of polynomial time black box polynomial identity testing (PIT), that have been hitherto solved using diverse tools and techniques, over fields of zero or large characteristic. In particular, we show that polynomial (in the size of the circuit) time hitting-set generators for identity testing of the two seemingly different and well studied models---depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas---can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence. By exploiting the Jacobian, we design the first efficient hitting-set generators for broad generalizations of the above-mentioned models, namely, (a) depth-3 ($\Sigma \Pi \Sigma$) circuits with constant transcendence degree of the polynomials computed by the product gates (no bounded top fanin restriction), and (b) constant-depth constant-occur formulas (no multilinear restriction). Constant occur ...