A tau-conjecture for Newton polygons

TL;DR

The paper introduces a tau-conjecture for Newton polygons of bivariate polynomials, linking their geometric complexity to computational complexity, and explores implications for circuit complexity and progress via combinatorial geometry.

Contribution

It proposes a new tau-conjecture for Newton polygons and shows its implications for the complexity of the permanent polynomial and related conjectures.

Findings

Weak form of the tau-conjecture implies permanent is not in P/poly

Progress made using recent combinatorial geometry results

Connections established between Newton polygons and circuit complexity

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.