Tangential center problem in dimension zero

Details Tangential center problem in dimension zero Tangential center problem in dimension zero

2012-02-27

arxiv.org/abs/1202.5896v1

Mathematics

Dynamical Systems

24 pages, 4 figures

Scientific paper

This paper is dedicated to the solution of the tangential center problem on 0-cycles: Given a polynomial $f\in\C[z]$ of degree $m$, denote $z_1(t),...,z_m(t)$ all algebraic functions defined by $f(z_k(t))=t$. Given integers $n_1,...,n_m$ such that $n_1+...+n_m=0$, find all polynomials $g\in\C[z]$ such that $n_1g(z_1(t))+...+n_mg(z_m(t))\equiv 0$. The classical Center-Focus Problem, or rather its tangential version in important non-trivial planar systems lead to the above problem. We prove 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 assumptions that in the above composition there is no merging of critical values. Under these assumptions, we give a complete (recursive) solution of the tangential center problem on 0-cycles. The recursive solution is obtained through three mechanisms: composition, primality and vanishing of the Newton-Girard component on projected cycles. The first mechanism was studied by Pakovich and Muzychuk and the authors among others in previous publications. The two later mechanisms are new.

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