4 research outputs found
Vanishing Abelian integrals on zero-dimensional cycles
In this paper we study conditions for the vanishing of Abelian integrals on
families of zero-dimensional cycles. That is, for any rational function ,
characterize all rational functions and zero-sum integers such
that the function vanishes identically. Here
are continuously depending roots of . We introduce a notion of
(un)balanced cycles. Our main result is an inductive solution of the problem of
vanishing of Abelian integrals when are polynomials on a family of
zero-dimensional cycles under the assumption that the family of cycles we
consider is unbalanced as well as all the cycles encountered in the inductive
process. We also solve the problem on some balanced cycles.
The main motivation for our study is the problem of vanishing of Abelian
integrals on single families of one-dimensional cycles. We show that our
problem and our main result are sufficiently rich to include some related
problems, as hyper-elliptic integrals on one-cycles, some applications to
slow-fast planar systems, and the polynomial (and trigonometric) moment problem
for Abel equation. This last problem was recently solved by Pakovich and
Muzychuk (\cite{PM} and \cite{P}). Our approach is largely inspired by their
work, thought we provide examples of vanishing Abelian integrals on zero-cycles
which are not given as a sum of composition terms contrary to the situation in
the solution of the polynomial moment problem.Comment: 35 pages, 1 figure; one reference added; abstract, introduction and
structure change
Unfoldings of saddle-nodes and their Dulac time
In this paper we study unfoldings of saddle-nodes and their Dulac time. By
unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem
A) we prove uniform regularity by which orbits and their derivatives arrive at
a node. Uniformity is with respect to all parameters including the unfolding
parameter bringing the node to a saddle-node and a parameter belonging to a
space of functions. In the second part, we apply this first result for proving
a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an
unfolding of a saddle-node. This result is a building block in the study of
bifurcations of critical periods in a neighbourhood of a polycycle. Finally, we
apply Theorems A and B to the study of critical periods of the Loud family of
quadratic centers and we prove that no bifurcation occurs for certain values of
the parameters (Theorem C)
Infinitesimal center problem on zero cycles and the composition conjecture
International audienceWe study the analog of the classical infinitesimal center problem in the plane, but for zero cycles. We define the displacement function in this context and prove that it is identically zero if and only if the deformation has a composition factor. That is, we prove that here the composition conjecture is true, in contrast with the tangential center problem on zero cycles. Finally, we give examples of applications of our results.Изучается аналог классической инфинитезимальной проблемы центра на плоскости для нулевых циклов. Для этого случая определяется функция смещения и доказывается, что она тождественно равна нулю тогда и только тогда, когда деформация имеет композиционный фактор. Иными словами, гипотеза композиции верна в этом случае, в отличие от тангенциальной проблемы центра для нулевых циклов. Приводятся примеры применения результатов