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Abstract:
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.
