  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.
