New Complexity Bounds for Certain Real Fewnomial Zero Sets

TL;DR

This paper establishes new bounds on the number of positive roots for certain bivariate polynomial systems and introduces explicit examples with more roots than previously known, advancing understanding of real fewnomial zero sets.

Contribution

It proves that systems with specific monomial counts can have more positive roots than previously documented and provides explicit examples and bounds on zero set complexity.

Findings

Systems with 3 and m monomials can have 2m-1 positive roots

Explicit example with m=4 having 7 positive roots

O(n^{11}) upper bound on diffeotopy types of zero sets

Abstract

Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m>=3, there are systems of the form (f,g) having exactly 2m-1 roots in the positive quadrant. Even examples with m=4 having 7 positive roots were unknown before this paper, so we detail an explicit example of this form. We also present an O(n^{11}) upper bound for the number of diffeotopy types of the real zero set of an n-variate polynomial with n+4 monomial terms.