Lower Bounds for the Circuit Size of Partially Homogeneous Polynomials

Abstract

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 λ (f ) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in P λ (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 λ (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 [12] which results in better lower bounds for circuit size (Lemma 6.7, Remark 6.8). Our methods yield non-trivial lower bound for the circuit size of several classes of multivariate homogeneous polynomials (Corollary 6.9, Example 6.10).