Sums of products of polynomials in few variables : lower bounds and polynomial identity testing

We study the complexity of representing polynomials as a sum of products of polynomials in few variables. More precisely, we study representations of the form $$P=\sum _{i=1}^T\prod _{j=1}^d{Q}_{ij}$$ such that each $Q_{ij}$ is an arbitrary polynomial that depends on at most $s$ variables. We prove the following results.

1. Over fields of characteristic zero, for every constant $\mu$ such that $0 \leq \mu < 1$,wegiveanexplicitfamilyofpolynomials${P_{N}}$,where$P_{N}$isofdegree$n$in$N = n^{O(1)}$ variables, such that any representation of the above type for $P_N$ with $s=N^{\mu }$ requires $Td \geq n^{\Omega(\sqrt{n})}$. This strengthens a recent result of Kayal and Saha [KS14a] which showed similar lower bounds for the model of sums of products of linear forms in few variables. It is known that any asymptotic improvement in the exponent of the lower bounds (even for $s=\sqrt{n}$) would separate VP and VNP[KS14a].
2. We obtain a deterministic subexponential time blackbox polynomial identity testing (PIT) algorithm for circuits computed by the above model when $T$ and the individual degree of each variable in $P$ are at most ${\log }^{O(1)}N$ and $s\le N^{\mu }$ for any constant $\mu <1\mathrm{/}2$. We get quasipolynomial running time when $s<{\log }^{O(1)}N$. The PIT algorithm is obtained by combining our lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04]. To the best of our knowledge, this is the first nontrivial PIT algorithm for this model (even for the case $s=2$), and the first nontrivial PIT algorithm obtained from lower bounds for small depth circuits.