The $16$th Hilbert problem on algebraic limit cycles

TL;DR

This paper proves a conjecture about the maximum number of algebraic limit cycles in certain polynomial vector fields, confirming a specific upper bound for systems with nodal invariant algebraic curves.

Contribution

It establishes the conjectured upper bound for algebraic limit cycles in polynomial vector fields with only nodal invariant algebraic curves, extending previous results.

Findings

Proved the conjecture for vector fields with only nodal invariant algebraic curves.

Confirmed the upper bound for algebraic limit cycles in these systems.

Extended the known results to a broader class of polynomial vector fields.

Abstract

For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree $m$?} In [J. Differential Equations, 248(2010), 1401--1409] Llibre, Ram\'irez and Sadovskia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: {\it Is $1+(m-1)(m-2)/2$ the maximal number of algebraic limit cycles that a polynomial vector field of degree $m$ can have?} In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre {\it et al}\,'s as a special one. For the polynomial vector fields having only non--dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.