On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of Valiant

TL;DR

This paper demonstrates, under GRH, the existence of complex polynomials with no small arithmetic circuits, and explores the relationship between Boolean and arithmetic circuit lower bounds in various characteristics.

Contribution

It establishes new lower bounds for uniform polynomials in VNP assuming GRH and links Boolean and arithmetic circuit complexity in characteristic zero and positive characteristic.

Findings

Existence of polynomials with no small arithmetic circuits under GRH.

Connection between Boolean and arithmetic circuit lower bounds.

Equivalence of fixed-polynomial size circuits for VNP polynomials and VP=VNP in positive characteristic.

Abstract

Assuming the Generalised Riemann Hypothesis (GRH), we show that for all k, there exist polynomials with coefficients in $\MA$ having no arithmetic circuits of size O(n^k) over the complex field (allowing any complex constant). We also build a family of polynomials that can be evaluated in AM having no arithmetic circuits of size O(n^k). Then we investigate the link between fixed-polynomial size circuit bounds in the Boolean and arithmetic settings. In characteristic zero, it is proved that $\NP \not\subset \size(n^k)$, or $\MA \subset \size(n^k)$, or NP=MA imply lower bounds on the circuit size of uniform polynomials in n variables from the class VNP over the complex field, assuming GRH. In positive characteristic p, uniform polynomials in VNP have circuits of fixed-polynomial size if and only if both VP=VNP over F_p and Mod_pP has circuits of fixed-polynomial size.