A tau-conjecture for Newton polygons

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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