Improved fewnomial upper bounds from Wronskians and dessins d'enfant

TL;DR

This paper introduces a novel approach combining dessins d'enfant and Wronskians to derive improved upper bounds on the number of positive solutions for certain fewnomial systems, advancing the understanding of polynomial root limits.

Contribution

It presents a new method using dessins d'enfant and Wronskians to establish tighter bounds on real solutions of fewnomial systems, improving previous polynomial bounds.

Findings

Bound of at most $ ext{deg} P + ext{deg} Q + 2$ roots in (0,1) for specific functions

New upper bounds on positive solutions for two-variable polynomial systems with few terms

Application of dessins d'enfant to real root counting in polynomial equations

Abstract

We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^\alpha(1-x)^{\beta} P - Q$, has at most $\deg P +\deg Q + 2$ roots in the interval $]0,~1[$. As a consequence, we obtain an upper bound on the number of positive solutions to a real polynomial system $f=g=0$ in two variables where $f$ has three monomials terms, and $g$ has $t$ terms. The approach we adopt for tackling this Fewnomial bound relies on the theory of Wronskians, which was used in Koiran et.\ al.\ (J.\ Symb.\ Comput., 2015) for producing the first upper bound which is polynomial in $t$.