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 \(a_i^2 > \tau a_{i-1}a_{i+1}\) for every \(i \in \{1,\ldots ,d-1\},\) where \(\tau > 0\). 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 \(d \leqslant k t^m\). Assuming that the \(f_{i,j}\) have only non-negative coefficients, we improve this degree bound to \(d = \mathcal O(k m^{2/3} t^{2m/3} \mathrm{log^{2/3}}(kt))\) if \(\tau > 1\), and to \(d \leqslant kmt\) 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 \(\mathsf {VP}\) and \(\mathsf {VNP}\). As they currently stand, these results are strong enough to provide a new example of a family of polynomials in \(\mathsf {VNP}\) which cannot be computed by monotone arithmetic circuits of polynomial size.