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

Authors Bruno Grenet, Pascal Koiran, Natacha Portier, Yann Strozecki



PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2011.127.pdf
  • Filesize: 454 kB
  • 13 pages

Document Identifiers

Author Details

Bruno Grenet
Pascal Koiran
Natacha Portier
Yann Strozecki

Cite As Get BibTex

Bruno Grenet, Pascal Koiran, Natacha Portier, and Yann Strozecki. The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 127-139, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011) https://doi.org/10.4230/LIPIcs.FSTTCS.2011.127

Abstract

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.

Subject Classification

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail