Log-concavity and lower bounds for arithmetic circuits

Abstract: One question that we investigate in this paper is, how can we build log-concave polynomials using sparse polynomials as building blocks? More precisely, let $f = \sum_{i = 0}^d a_i X^i \in \mathbb{R}^+[X]$ be a polynomial satisfying the log-concavity condition $a_i^2 \textgreater{} \tau a_{i-1}a_{i+1}$ for every $i \in {1,\ldots,d-1},$ where $\tau \textgreater{} 0$. Whenever $f$ can be written under the form $f = \sum_{i = 1}^k \prod_{j = 1}^m f_{i,j}$ where the polynomials $f_{i,j}$ have at most $t$ monomials, it is clear that $d \leq k t^m$. Assuming that the $f_{i,j}$ have only non-negative coefficients, we improve this degree bound to $d = \mathcal O(k m^{2/3} t^{2m/3} {\rm log^{2/3}}(kt))$ if $\tau \textgreater{} 1$, and to $d \leq kmt$ if $\tau = d^{2d}$. This investigation has a complexity-theoretic motivation: we show that a suitable strengthening of the above results would imply a separation of the algebraic complexity classes VP and VNP. As they currently stand, these results are strong enough to provide a new example of a family of polynomials in VNP which cannot be computed by monotone arithmetic circuits of polynomial size.