A tau-conjecture for Newton polygons
Abstract
One can associate to any bivariate polynomial P(X,Y) its Newton polygon. This is the convex hull of the points (i,j) such that the monomial X^i Y^j appears in P with a nonzero coefficient. We conjecture that when P is expressed as a sum of products of sparse polynomials, the number of edges of its Newton polygon is polynomially bounded in the size of such an expression. We show that this "tau-conjecture for Newton polygons," even in a weak form, implies that the permanent polynomial is not computable by polynomial size arithmetic circuits. We make the same observation for a weak version of an earlier "real tau-conjecture." Finally, we make some progress toward the tau-conjecture for Newton polygons using recent results from combinatorial geometry.
Cite
@article{arxiv.1308.2286,
title = {A tau-conjecture for Newton polygons},
author = {Pascal Koiran and Natacha Portier and Sébastien Tavenas and Stéphan Thomassé},
journal= {arXiv preprint arXiv:1308.2286},
year = {2014}
}