# The 16th Hilbert problem restricted to circular algebraic limit cycles

Jaume Llibre, Rafael Ramírez, Valentín Ramírez, Natalia Sadovskaia

Research output: Contribution to journalArticleResearchpeer-review

12 Citations (Scopus)

## Abstract

© 2015 Elsevier Inc. We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S-1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S-1 algebraic limit cycles given by circles, and this number is reached.
Original language English 5726-5760 Journal of Differential Equations 260 7 https://doi.org/10.1016/j.jde.2015.12.019 Published - 5 Apr 2016

## Keywords

• 16th Hilbert's problem
• Algebraic limit circles
• Darboux integrability
• Invariant algebraic circles
• Planar polynomial differential system
• Polynomial vector fields