The 16th Hilbert problem on algebraic limit cycles

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:251

Article

Zhang, Xiang

关键词: Polynomial differential systems;    Holomorphic singular foliations;    Simple version of the 16th Hilbert problem;    Algebraic limit cycles;   

DOI  :  10.1016/j.jde.2011.06.008

来源: Elsevier

【 摘 要 】

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramirez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramirez and Sadovskaia 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: 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 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. (C) 2011 Elsevier Inc. All rights reserved.

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