利用嵌套Artin逼近的生成函数代数性
The algebraicity of generating functions using nested Artin approximation
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中文总结 AI 辅助
本文给出二元嵌套Artin-Popescu逼近定理的初等证明，用于确保多项式方程存在嵌套代数幂级数解，从而简化枚举组合学中生成函数代数性的证明。
中文摘要 AI 辅助
我们给出了二元嵌套Artin-Popescu逼近定理的一个初等证明：该定理确保给定多项式（函数）方程存在嵌套代数幂级数解。这类方程经常出现在枚举组合学中，例如在第一象限计数格路时。因此，Bousquet-Mélou和Jehanne提出的生成函数代数性的特设证明可以被应用该定理所取代，而无需诉诸于Popescu解决的极其困难的一般情形。我们的证明方法紧密遵循Denef和Lipshitz的论证，他们处理了更一般的情形（即Weierstrass系统的Artin逼近）。这结合了Hauser和Woblistin用于描述由所有幂级数解形成的无限维簇的整体几何的技术。将这两种方法放在一起，现在为组合学家提供了二元嵌套Artin-Popescu逼近定理的一个可理解的证明。
英文摘要
We present an elementary proof of the bivariate nested Artin-Popescu approximation theorem: it ensures the existence of a nested algebraic power series solution of a given polynomial (functional) equation. Such equations often appear in enumerative combinatorics, e.g., when counting lattice walks in the first quadrant. The ad hoc proofs of the algebraicity of the generating functions, as e.g. proposed by Bousquet-Mélou and Jehanne , can thus be replaced by applying the theorem without need to resort to the extremely difficult general case solved by Popescu. Our method of proof follows closely the arguments of Denef and Lipshitz who treat a more general case (namely, the Artin approximation for Weierstrass systems). This is combined with the techniques of Hauser and Woblistin for the description of the overall geometry of the infinite dimensional variety formed by all power series solutions. Putting both approaches together now provides combinatorialists with an accessible proof for the bivariate nested Artin-Popescu approximation theorem.
