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Abderaouf Mourtada

Publications and source records attributed to Abderaouf Mourtada.

2 recordsLinked to original sources
arXiv2009

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

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.

math.DS↗
arXiv2009

Extension aux cycles singuliers du theoreme de Khovanski-Varchenko

Let dH be a Hamiltonian one form on the real plane, of degre d. We show that, if H is a Morse function, generic at infinity, then there exists a number N(d) depending only on d, such that every small perturbation of dH has at most N(d) limit cycles on the hole real plane, assuming that it's of degre at most d, and that it has a non vanishing Abelian integral along real cycles of dH.

math.DS↗