Action de derivations irreductibles sur les algebres quasi-regulieres d'Hilbert

TL;DR

This paper investigates the action of irreducible derivations on quasi-regular Hilbert algebras, establishing finiteness properties and applying these results to prove a significant case of Hilbert's 16th problem regarding limit cycles.

Contribution

It introduces new finiteness results for derivations acting on Hilbert's quasi-regular algebras and applies these to limit cycle accumulation in vector field families.

Findings

Algebras are X-finite or locally X-finite.

Differential ideals are noetherian or locally noetherian.

No accumulation of limit cycles on hyperbolic polycycles.

Abstract

We study the action of irreducible derivations X on some Hilbert's quasi-regular algebras QRH of germes at 0 of analytic functions on (U,0), where U is a semi-algebraic set: that is, we show that these algebras are X-finite or locally X-finite, ie. the degre of the integral projection is finite by restriction to fibers of elements of QRH, and the differential ideals are noetherian or locally noetherian. We then give an important application of this material to the Hilbert's 16th problem about limit cycles: there is no accumulation of limit cycles on hyperbolic polycycles, inside compact analytic families of vector fields on the 2-sphere. This is a highly non trivial result as it includes the case of polycycle that is an accumulation of cycles.