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 $P_{\lambda }(f)$ 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}).