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.
Document type :
Reports
Complete list of metadatas

Cited literature [28 references]  Display  Hide  Download

https://hal-ens-lyon.archives-ouvertes.fr/ensl-00850791
Contributor : Pascal Koiran <>
Submitted on : Monday, May 12, 2014 - 8:51:02 PM
Last modification on : Thursday, February 7, 2019 - 4:52:46 PM
Long-term archiving on : Tuesday, August 12, 2014 - 12:20:30 PM

Files

newton.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : ensl-00850791, version 2
  • ARXIV : 1308.2286

Collections

Citation

Pascal Koiran, Natacha Portier, Sébastien Tavenas, Stéphan Thomassé. A tau-conjecture for Newton polygons. 2014, pp.14. ⟨ensl-00850791v2⟩

Share

Metrics

Record views

237

Files downloads

158