Counting Real Connected Components of Trinomial Curve Intersections and m-nomial Hypersurfaces

TL;DR

This paper establishes sharp upper bounds on the number of real roots and connected components of fewnomial systems, significantly improving previous bounds and extending to systems with real exponents and degeneracies.

Contribution

It proves the first sharp bounds for the number of positive roots of bivariate trinomials and extends these bounds to certain n-variate systems, surpassing earlier exponential bounds.

Findings

Maximum of 5 isolated positive roots for bivariate trinomials

Improved bounds on the number of connected components of real zero sets

Extensions to systems with real exponents and degeneracies

Abstract

We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.