A sharp bound on the number of real intersection points of a sparse plane curve with a line

TL;DR

This paper establishes a sharp upper bound of 6t - 7 on the number of real intersection points between a line and a plane curve defined by a polynomial with at most t monomials, improving previous results.

Contribution

It provides a new tight bound for the intersection points of sparse polynomials with lines, using advanced algebraic geometry techniques like Grothendieck's dessins d'enfant.

Findings

Bound of 6t - 7 for t ≥ 3 is sharp

The bound applies to real intersection points, excluding infinity

Improves previous bounds by M. Avendano

Abstract

We prove that the number of real intersection points of a real line with a real plane curve defined by a polynomial with at most t monomials is either infinite or does not exceed 6t -7. This improves a result by M. Avendano. Furthermore, we prove that this bound is sharp for t = 3 with the help of Grothendieck's dessins d'enfant.