Grenet, Bruno ;
Koiran, Pascal ;
Portier, Natacha ;
Strozecki, Yann
The Limited Power of Powering: Polynomial Identity Testing and a Depthfour Lower Bound for the Permanent
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 tauconjecture" which is inspired by this connection.
The real tauconjecture 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 tauconjecture holds true for a restricted class of sums of products of sparse polynomials.
This result yields lower bounds for a restricted class of depth4 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.
BibTeX  Entry
@InProceedings{grenet_et_al:LIPIcs:2011:3350,
author = {Bruno Grenet and Pascal Koiran and Natacha Portier and Yann Strozecki},
title = {{The Limited Power of Powering: Polynomial Identity Testing and a Depthfour Lower Bound for the Permanent}},
booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)},
pages = {127139},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897347},
ISSN = {18688969},
year = {2011},
volume = {13},
editor = {Supratik Chakraborty and Amit Kumar},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3350},
URN = {urn:nbn:de:0030drops33501},
doi = {http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2011.127},
annote = {Keywords: Algebraic Complexity, Sparse Polynomials, Descartes' Rule of Signs, Lower Bound for the Permanent, Polynomial Identity Testing}
}
2011
Keywords: 

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

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

Related Scholarly Article: 


Issue date: 

2011 
Date of publication: 

2011 