Lower Bounds for the Circuit Size of Partially Homogeneous Polynomials

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_⋋(f) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in P_⋋(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_⋋(f). We discuss some algebraic methods for proving the (s, r)-(weak) elusiveness. We also improve estimates for the normal-homogeneous form of an arithmetic circuit obtained by Raz, which results in better lower bounds for circuit size. Our methods yield nontrivial lower bound for the circuit size of several classes of multivariate homogeneous polynomials.