Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion

Kabanets and Impagliazzo cite{KaIm04} show how to decide the circuit polynomial identity testing problem (CPIT) in deterministic subexponential time, assuming hardness of some explicit multilinear polynomial family ${f_m}_{m geq 1}$ for arithmetic circuits. In this paper, a special case of CPIT is considered, namely non-singular matrix completion ($NSMC$) under a low-individual-degree promise. For this subclass of problems it is shown how to obtain the same deterministic time bound, using a weaker assumption in terms of the {em determinantal complexity} $dcomp(f_m)$ of $f_m$. Building on work by Agrawal cite{Agr05}, hardness-randomness tradeoffs will also be shown in the converse direction, in an effort to make progress on Valiant\u27s $VP$ versus $VNP$ problem. To separate $VP$ and $VNP$, it is known to be sufficient to prove that the determinantal complexity of the $mtimes m$ permanent is $m^{omega(log m)}$. In this paper it is shown, for an appropriate notion of explicitness, that the existence of an explicit multilinear polynomial family ${f_m}_{m geq 1}$ with $dcomp(f_m) = m^{omega(log m)}$ is equivalent to the existence of an efficiently computable {em generator} ${G_n}_{ngeq 1}$ {em for} multilinear $NSMC$ with seed length $O(n^{1/sqrt{log n}})$. The latter is a combinatorial object that provides an efficient deterministic black-box algorithm for $NSMC$. ``Multilinear $NSMC$\u27\u27 indicates that $G_n$ only has to work for matrices $M(x)$ of $poly(n)$ size in $n$ variables, for which $det(M(x))$ is a multilinear polynomial