RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mosc. Math. J.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mosc. Math. J., 2013, Volume 13, Number 4, Pages 555–583 (Mi mmj504)  

Inductive solution of the tangential center problem on zero-cycles

A. Álvareza, J. L. Bravoa, P. Mardešićb

a Departamento de Matemáticas, Universidad de Extremadura, Avenida de Elvas s/n, 06006 Badajoz Spain
b Université de Bourgogne, Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, UFR Sciences et Techniques, 9, av. A. Savary, BP 47870, 21078 Dijon Cedex, France

Abstract: Given a polynomial $f\in\mathbb C[z]$ of degree $m$, let $z_1(t),…,z_m(t)$ denote all algebraic functions defined by $f(z_k(t))=t$. Given integers $n_1,…,n_m$ such that $n_1+…+n_m=0$, the tangential center problem on zero-cycles asks to find all polynomials $g\in\mathbb C[z]$ such that $n_1g(z_1(t))+…+n_mg(z_m(t))\equiv0$. The classical Center-Focus Problem, or rather its tangential version in important non-trivial planar systems lead to the above problem.
The tangential center problem on zero-cycles was recently solved in a preprint by Gavrilov and Pakovich.
Here we give an alternative solution based on induction on the number of composition factors of $f$ under a generic hypothesis on $f$. First we show the uniqueness of decompositions $f=f_1\circ…\circ f_d$ such that every $f_k$ is $2$-transitive, monomial or a Chebyshev polynomial under the assumption that in the above composition there is no merging of critical values.
Under this assumption, we give a complete (inductive) solution of the tangential center problem on zero-cycles. The inductive solution is obtained through three mechanisms: composition, primality and vanishing of the Newton-Girard component on projected cycles.

Key words and phrases: Abelian integrals, tangential center problem, center-focus problem, moment problem.

DOI: https://doi.org/10.17323/1609-4514-2013-13-4-555-583

Full text: http://www.mathjournals.org/.../2013-013-004-001.html
References: PDF file   HTML file

Bibliographic databases:

MSC: 34C07, 34C08, 34M35, 14K20
Received: April 18, 2012; in revised form December 10, 2012
Language:

Citation: A. Álvarez, J. L. Bravo, P. Mardešić, “Inductive solution of the tangential center problem on zero-cycles”, Mosc. Math. J., 13:4 (2013), 555–583

Citation in format AMSBIB
\Bibitem{AlvBraMar13}
\by A.~\'Alvarez, J.~L.~Bravo, P.~Marde{\v s}i{\'c}
\paper Inductive solution of the tangential center problem on zero-cycles
\jour Mosc. Math.~J.
\yr 2013
\vol 13
\issue 4
\pages 555--583
\mathnet{http://mi.mathnet.ru/mmj504}
\crossref{https://doi.org/10.17323/1609-4514-2013-13-4-555-583}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3184072}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000330037700001}


Linking options:
  • http://mi.mathnet.ru/eng/mmj504
  • http://mi.mathnet.ru/eng/mmj/v13/i4/p555

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Moscow Mathematical Journal
    Number of views:
    This page:75
    References:33

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020