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A tau-conjecture for Newton polygons

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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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Dates and versions

ensl-00850791 , version 1 (09-08-2013)
ensl-00850791 , version 2 (12-05-2014)

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