Jacobian Hits Circuits: Hitting-Sets, Lower Bounds for Depth-{D} {Occur-k} Formulas \& Depth-3 Transcendence {Degree-k} Circuits

We present a single common tool to strictly subsume \emphall} known cases of polynomial time blackbox 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 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: \emph{Jacobian} captures algebraic independence. By exploiting the Jacobian, we design the {\em first} efficient hitting-set generators for broad generalizations of the bove-mentioned models, namely: \begin{itemize} \item depth-3 (Σ \Pi Σ) circuits with constant \emph{transcendence degree} of the polynomials computed by the product gates (\emph{no} bounded top fanin restriction), and \item constant-depth constant-\emph{occur} formulas (\emph{no} multilinear restriction). \end{itemize} Constant-\emph{occur} of a variable, as we define it, is a much more general concept than constant-read. Also, earlier work on the latter model assumed that the formula is multilinear. Thus, our work goes further beyond the related results obtained by Saxena & Seshadhri (STOC 2011), Saraf & Volkovich (STOC 2011), Anderson et al.\ (CCC 2011), Beecken et al.\ (ICALP 2011) and Grenet et al.\ (FSTTCS 2011), and brings them under one unifying technique. In addition, using the same Jacobian based approach, we prove exponential lower bounds for the immanant (which includes permanent and determinant) on the \emph{same depth-3 and depth-4 models for which we give efficient PIT algorithms. Our results reinforce the intimate connection between identity testing and lower bounds by exhibiting a concrete mathematical tool - the Jacobian - that is equally effective in solving both the problems on certain interesting and previously well-investigated (but not well understood) models of computation