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On the number of algebraically independent Poincaré-Liapunov constants. (English) Zbl 1124.34018

The center-focus problem is the problem of determining whether the isolated singularity at the origin of the planar system of ordinary differential equations

$\stackrel{˙}{x}=-y+P\left(x,y\right),\phantom{\rule{2.em}{0ex}}\stackrel{˙}{y}=x+Q\left(x,y\right)$

(where $P$ and $Q$ are polynomial functions without constant or linear terms) is a center (all trajectories in a punctured neighborhood of the origin are ovals) or a focus (all such trajectories are spirals). The cyclicity problem is to determine the maximum number of limit cycles that can be made to bifurcate from the origin under small perturbation within the set of polynomial systems of some fixed degree. The Poincaré-Lyapunov Theorem asserts that the origin is a center if and only if there is a formal first integral of the form

$H\left(x,y\right)=\frac{1}{2}\left({x}^{2}+{y}^{2}\right)=\cdots ·$

More generally one looks for a formal power series of this form for which

$\stackrel{˙}{H}\left(x,y\right)=\sum _{k=2}^{\infty }{V}_{2k}{\left({x}^{2}+{y}^{2}\right)}^{k};$

the ${V}_{2k}$ are polynomials in the coefficients of $P$ and $Q$, traditionally called the Poincaré-Lyapunov quantities. The cyclicity problem can be addresssed by investigating the ideal generated by the Poincaré-Lyapunov quantities in the corresponding polynomial ring. After preliminaries along these lines and the change to polar coordinates, the author proves several theorems concerning the number of independent Poincaré-Lyapunov quantities and makes conjectures relating to this number and whether the ideals under consideration must be radical (giving additional evidence that they are).

##### MSC:
 34C07 Theory of limit cycles of polynomial and analytic vector fields 34C05 Location of integral curves, singular points, limit cycles (ODE)
##### Keywords:
Poincaré-Lyapunov quantity; limit cycle