Dense Fewnomials

TL;DR

This paper introduces new bounds on the number of real solutions for polynomial systems with intermediate structure between sparse and dense, using Gale duality and stratified Morse theory, generalizing previous results.

Contribution

It develops a modified Gale duality approach and stratified Morse theory to derive bounds for dense fewnomials, extending prior bounds for ordinary fewnomials.

Findings

Derived new bounds for real solutions of structured polynomial systems

Generalized previous bounds for ordinary fewnomials

Bound the Betti number of hypersurfaces defined by dense fewnomials

Abstract

We derive new bounds of fewnomial type for the number of real solutions to systems of polynomials that have structure intermediate between fewnomials and generic (dense) polynomials. This uses a modified version of Gale duality for polynomial systems. We also use stratified Morse theory to bound the total Betti number of a hypersurface defined by such a dense fewnomial. These bounds contain and generalize previous bounds for ordinary fewnomials obtained by Bates, Bertrand, Bihan, and Sottile.