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 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}). }