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.
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https://hal-ens-lyon.archives-ouvertes.fr/ensl-00850791
Contributor : Pascal Koiran <>
Submitted on : Friday, August 9, 2013 - 4:53:14 PM
Last modification on : Friday, January 25, 2019 - 3:34:10 PM
Long-term archiving on : Sunday, November 10, 2013 - 3:00:15 AM

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  • HAL Id : ensl-00850791, version 1
  • ARXIV : 1308.2286

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Pascal Koiran, Natacha Portier, Sébastien Tavenas, Stéphan Thomassé. A tau-conjecture for Newton polygons. 2013, pp.13. ⟨ensl-00850791v1⟩

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