Lower bounds for the circuit size of partially homogeneous polynomials

Hông Vân Lê

doi:10.1007/s10958-017-3483-4

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arXiv · 1302.3360

Lower bounds for the circuit size of partially homogeneous polynomials

Hông Vân Lê

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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 \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}).

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Hông Vân Lê. 2015-03-16. Lower bounds for the circuit size of partially homogeneous polynomials. https://doi.org/10.1007/s10958-017-3483-4

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