Counting positive intersection points of a trinomial and a $\mathbf{T}$-nomial curves via Groethendieck's dessin d'enfant

TL;DR

This paper establishes a new upper bound on the number of positive solutions for certain bivariate polynomial systems using Groethendieck's dessin d'enfant, improving previous bounds and providing detailed geometric and combinatorial insights.

Contribution

It introduces a novel application of Groethendieck's dessin d'enfant to bound positive solutions of polynomial systems, refining existing bounds and characterizing extremal cases.

Findings

New upper bound: 3·2^{t-2} - 1 for positive solutions.

Refinement of previous bounds for t=4,...,9.

Characterization of systems reaching the bound for t=3.

Abstract

We consider real polynomial systems $f=g=0$ in two variables where $f$ has $t\geq 3$ monomial terms and $g$ has $3$ monomials terms. We prove that the number of positive isolated solutions of such a system does not exceed $3\cdot 2^{t-2} - 1$. This improves the bound $2^t - 2$ obtained in [T.-Y. Li, J.-M. Rojas and X. Wang, 2003]. This also refines for $t=4,\ldots,9$ the bound $2t^3/3 + 5t$ obtained in [P. Koiran, N. Portier and S. Tavenas, 2015]. Our proof is based on a delicate analysis of the Groethendieck's dessin d'enfant associated to some rational function determined by the system. For $t=3$, it was shown in [T.-Y. Li, J.-M. Rojas and X. Wang, 2003] that the sharp bound is five, and if this bound is reached, then the Minkowski sum of the associated Newton triangles is an hexagon. A further analysis of Groethendieck's dessin d'enfant allows us to show that if the bound five is reached, then there exist two consecutive edges of the hexagon which are translate of two consecutive edges of one Newton triangle.