The 16th Hilbert problem restricted to circular algebraic limit cycles

Abstract

Agraïments: FEDER-UNAB10-4E-378 and Consolider CSD2007-00004 "ES"

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.