Vanishing Abelian integrals on zero-dimensional cycles

Details Vanishing Abelian integrals on zero-dimensional cycles Vanishing Abelian integrals on zero-dimensional cycles

2011-01-10

arxiv.org/abs/1101.1777v2

Mathematics

Dynamical Systems

35 pages, 1 figure; one reference added; abstract, introduction and structure changed

Scientific paper

In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function $f(z)$, characterize all rational functions $g(z)$ and zero-sum integers $\{n_i\}$ such that the function $t\mapsto\sum n_ig(z_i(t))$ vanishes identically. Here $z_i(t)$ are continuously depending roots of $f(z)-t$. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when $f,g$ 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.

Affiliated with

Also associated with

No associations

Rating

Vanishing Abelian integrals on zero-dimensional cycles does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Vanishing Abelian integrals on zero-dimensional cycles, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Vanishing Abelian integrals on zero-dimensional cycles will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-456265