Log-concavity and lower bounds for arithmetic circuits

TL;DR

This paper explores the construction of log-concave polynomials from sparse polynomials, providing new degree bounds and implications for algebraic complexity classes, including VP and VNP.

Contribution

It introduces improved degree bounds for log-concave polynomials expressed as sums of products of sparse polynomials and links these bounds to complexity class separations.

Findings

Improved degree bounds for log-concave polynomials in terms of sparsity and structure.

Demonstrates that certain bounds imply VP vs. VNP separation.

Provides examples of polynomials in VNP not computable by small monotone 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.