

(LaTeX) Abstract: 
We present a single common tool to strictly subsume \emph{all} 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 hittingset generators for identity testing of the two seemingly different and
well studied models  depth$3$ circuits with bounded top fanin, and constantdepth constantread multilinear formulas  can be constructed using one common algebraicgeometry theme: \emph{Jacobian} captures algebraic independence. By exploiting the Jacobian, we design the {\em first} efficient hittingset generators for broad generalizations of the bovementioned models, namely: \begin{itemize} \item depth$3$ ($\Sigma \Pi \Sigma$) circuits with constant \emph{transcendence degree} of the polynomials computed by the
product gates (\emph{no} bounded top fanin restriction), and \item constantdepth 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 constantread. 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 wellinvestigated (but not well understood) models of computation. 




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