Lower bounds for the circuit size of partially homogeneous polynomials

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 Pλ(f) of m-tuples of homogeneous polynomials of equal degree such that the circuit size of any member in Pλ(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 Pλ(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). To my Teacher Anatoly Timofeevich Fomenko