The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

Bruno Grenet (1, 2) , Pascal Koiran (1, 2) , Natacha Portier (1, 2) , Yann Strozecki (2, 3)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 DCS - Department of Computer Science [University of Toronto]
3 ELM - Équipe de Logique Mathématique

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real tau-conjecture" which is inspired by this connection. The real tau-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real tau-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.

Keywords: Algebraic Complexity; Sparse Polynomials; Descartes' Rule of Signs; Lower Bound for the Permanent; Polynomial Identity Testing

Domains: Computer Science [cs]; Computational Complexity [cs.CC]

Bruno Grenet, Pascal Koiran, Natacha Portier, Yann Strozecki. The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'11), Indian Association for Research in Computing Science, Dec 2011, Mumbai, India. pp.16, ⟨10.4230/LIPIcs.FSTTCS.2011.127⟩. ⟨ensl-00607154⟩