On a variant of Hilbert's 16th problem

TL;DR

This paper investigates the relationship between the number of monomials in planar polynomial vector fields and the maximum number of limit cycles, establishing quadratic growth and providing bounds for small m.

Contribution

It demonstrates that the number of limit cycles grows at least quadratically with the number of monomials and offers specific lower bounds for fields with up to 10 monomials.

Findings

Limit cycles increase at least quadratically with the number of monomials.

Provided explicit lower bounds for systems with up to 10 monomials.

Established growth rate of limit cycles as a function of monomials.

Abstract

We study the number of limit cycles that a planar polynomial vector field can have as a function of its number $m$ of monomials. We prove that the number of limit cycles increases at least quadratically with $m$ and we provide good lower bounds for $m\leqslant10$.