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 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

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
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Bibliographic databases:

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