A tau-conjecture for Newton polygons

One can associate to any bivariate polynomial $$P(X,Y)$$P(X,Y) its Newton polygon. This is the convex hull of the points $$(i,j)$$(i,j) such that the monomial $$X^i Y^j$$XiYj appears in $$P$$P with a nonzero coefficient. We conjecture that when $$P$$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 $$\tau $$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 $$\tau $$τ-conjecture. Finally, we make some progress toward the $$\tau $$τ-conjecture for Newton polygons using recent results from combinatorial geometry.