Lower bounds for the circuit size of partially homogeneous polynomials

TL;DR

This paper introduces a new approach to establish lower bounds on the circuit size of partially homogeneous polynomials by associating polynomial families and analyzing their elusiveness, improving previous estimates.

Contribution

It develops a method linking polynomial elusiveness to circuit size lower bounds and improves existing estimates for homogeneous-form circuits.

Findings

Provides a framework for lower bounds via (s,r)-elusiveness

Improves estimates in Raz's homogeneous-form circuits

Achieves non-trivial lower bounds for certain polynomial classes

Abstract

In this paper we associate to each multivariate polynomial $f$ that is homogeneous relative to a subset of its variables a series of polynomial families $P_\lambda (f)$ of $m$-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in $P_\lambda (f)$ is bounded from above by the circuit size of $f$. This provides a method for obtaining lower bounds for the circuit size of $f$ by proving $(s,r)$-(weak) elusiveness of the polynomial mapping associated with $P_\lambda (f)$. We discuss some algebraic methods for proving the $(s,r)$-(weak) elusiveness. We also improve estimates in the normal homogeneous-form of an arithmetic circuit obtained by Raz in \cite{Raz2009} which results in better lower bounds for circuit size (Lemma \ref{lem:cor38}, Remark \ref{rem:cor38}). Our methods yield non-trivial lower bound for the circuit size of several classes of multivariate homogeneous polynomials (Corollary \ref{cor:412}, Example \ref{ex:bi}).