零圈切心问题的归纳解
Inductive Solution of the Tangential Center Problem on Zero-Cycles
摘要：
给定次数为$m$的多项式$f\in\C[z]$，令$z_1(t),...,z_m(t)$表示由$f(z_k(t))=t$定义的所有代数函数。给定整数$n_1...,n_m$使得$n_1+...+n_m=0$，零圈上的切向中心问题要求找到所有多项式$g\in\C[z]$使得$n_1g(z_1(t))+...+n_mg(z_m(t))\equiv 0$。经典的中心焦点问题，或者更确切地说，它的切向版本在重要的非平凡的平面系统导致上述问题。零圈的切向中心问题最近在Gavrilov和Pakovich的预印本中得到了解决。在这里，我们给出了另一种解决方案的基础上归纳$f$的组成因素的数量下的一个通用的假设$f$。首先，我们证明了分解的唯一性$f=f_1\circ...\circ f_d$，使每一个$f_k$是2-传递，单项式或切比雪夫多项式的假设下，在上述组合没有合并的临界值。在此假设下，我们给出了零圈切中心问题的完整（归纳）解。归纳的解决方案是通过三个机制：合成，素性和消失的牛顿-吉拉德分量的投影圈。
英文摘要：
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.
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