Seminar za dinamičke sustave

PMF Matematički odsjek
22.02.2022 - 16:15 - 18:00
On Tuesday, 22/02/2022, at 16.15, in  room A101 at Department of Mathematics, Zagreb,
Pavao MardešićUniversity of Burgundy, 
will give a talk in Dynamical Systems Seminar under the title: 
Darboux relative exactness.
The abstract can be found below in the mail.
The talk will be in English.
Everybody is invited. 




I will present a joint work with Colin Christopher.
We study polynomial forms $\omega$ in the plane having a family of invariant cycles $\gamma(t)$ and a Darboux first integral i.e. a first integral of the form $F(x,y)=\prod f_i^{\lambda_i}(x,y)$, with $f_i$ polynomials and $\lambdi>0$.
We study polynomial deformations $\omega+\epsilon\eta$ of this form and the corresponding displacement function $\Delta$ along the cycles $\gamma$. The displacement function $\Delta$ is of the form
\Delta_\epsilon(t)=\epsilon M_1(t)+o(\epsilon),$$
where the function $M_1(t)$ is a pseudo-abelian integral (i.e. integral of a rational form along the cycle $\gamma(t)$).
Generalizing Ilyashenko’s work we introduce the notion of \emph{Darboux relative exactness} and show that under generic hypothesis on the first integral $F$, the  leading term $M_1$ is identically equal to zero if and only if the form $\eta$ is Darboux relatively exact.

We also give some corollaries of this result.



Best regards,

Maja Resman


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