Inductive Solution of the Tangential Center Problem on Zero-Cycles

TL;DR

This paper presents an inductive method to solve the tangential center problem on zero-cycles for polynomials, providing a complete solution under generic conditions by analyzing polynomial decompositions and their properties.

Contribution

It introduces an alternative inductive approach to solving the tangential center problem, focusing on polynomial decompositions and generic hypotheses, expanding on prior solutions.

Findings

Unique decomposition of polynomials into 2-transitive, monomial, or Chebyshev factors.

Complete inductive solution under generic conditions.

Solution utilizes composition, primality, and Newton-Girard components.

Abstract

Given a polynomial $f\in\C[z]$ 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 $g\in\C[z]$ 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. 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.