A τ-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 so-called τ -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 τ -conjecture. Finally, we make some progress toward the τ -conjecture for Newton polygons using recent results from combinatorial geometry.