Taylor控制、差分方程和Bautin理想
Taylor Domination, Difference Equations, and Bautin Ideals
摘要：
我们比较了三种方法来研究的解析函数$f(z)=\sum_{k=0}^\infty a_kz^k$的泰勒系数的行为。第一种是$f(z)$在复圆盘$D_R$中的“泰勒支配”性质，它是一个形式为\[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \]的不等式。第二种方法是基于通过递归关系生成$a_k$的可能性。具体来说，我们考虑线性非平稳递归的形式\[ a_{k}=\sum_{j=1}^{d}c_{j}(k)\cdot a_{k-j},\ \ k=d,d+1,\dots, \]一致有界系数。在第三种方法中，我们假设$a_k=a_k(λ)$是有限维参数$λ\in {\mathbb C}^n.$中的多项式。我们研究了$λ$中多项式环${\mathbb C}[λ]$中由$a_{1}(λ),\ldots,a_{k}(λ)$生成的“Bautin理想”$I_k$。\smallskip这三种方法被证明是密切相关的。我们在这个方向提出了一些结果和问题。
英文摘要：
We compare three approaches to studying the behavior of an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ from its Taylor coefficients. The first is "Taylor domination" property for $f(z)$ in the complex disk $D_R$, which is an inequality of the form \[ |a_{k}|R^{k}\leq C\ \max_{i=0,\dots,N}\ |a_{i}|R^{i}, \ k \geq N+1. \] The second approach is based on a possibility to generate $a_k$ via recurrence relations. Specifically, we consider linear non-stationary recurrences of the form \[ a_{k}=\sum_{j=1}^{d}c_{j}(k)\cdot a_{k-j},\ \ k=d,d+1,\dots, \] with uniformly bounded coefficients. In the third approach we assume that $a_k=a_k(λ)$ are polynomials in a finite-dimensional parameter $λ\in {\mathbb C}^n.$ We study "Bautin ideals" $I_k$ generated by $a_{1}(λ),\ldots,a_{k}(λ)$ in the ring ${\mathbb C}[λ]$ of polynomials in $λ$. \smallskip These three approaches turn out to be closely related. We present some results and questions in this direction.
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