论文标题
Pólya-Carlson二分法和应用的一般标准
A general criterion for the Pólya-Carlson dichotomy and application
论文作者
论文摘要
我们证明了非理性功率系列$ f（z）= \ displayStyle \ sum_ {n = 0}^{\ infty} a_nz^n $具有数字字段$ k $中的系数，以将单位圈作为自然边界接收到单位圈子。作为一个应用程序，让$ f $为有限的字段，让$ d $成为一个积极的整数，让$ a \ in m_d（f [t]）$为a $ d \ times d $ -matrix，带有$ f [t] $中的条目，让$ f [t] $，让$ζ_A（z）$与Artin-Mazur Zeta函数相关联，与Artin-Mazur Zeta函数相关联$ $ $ $ MAP ON COMPACT-$ A $ A $ A $ A $ A $ A $ A $ A $ A $ A $ A $ A $ a $ $ f（（1/t））^d/f [t]^d $。我们提供了$ζ_A（z）$为代数时的完整表征，并证明它在超越情况下将收敛圈作为自然边界。这与$ \ mathbb {r}^d/\ mathbb {z}^d $上的线性内态形态形成鲜明对比，其中Baake，Lau和Paskunas证明Zeta函数始终是合理的。讨论了与贝尔，Byszewski，Cornelissen，Miles，Royals和Ward的早期作品的一些联系。我们的方法在贝尔，Nguyen和Zannier的最新工作中使用了类似的技术，以及涉及线性复发序列的某些修补参数。
We prove a general criterion for an irrational power series $f(z)=\displaystyle\sum_{n=0}^{\infty}a_nz^n$ with coefficients in a number field $K$ to admit the unit circle as a natural boundary. As an application, let $F$ be a finite field, let $d$ be a positive integer, let $A\in M_d(F[t])$ be a $d\times d$-matrix with entries in $F[t]$, and let $ζ_A(z)$ be the Artin-Mazur zeta function associated to the multiplication-by-$A$ map on the compact abelian group $F((1/t))^d/F[t]^d$. We provide a complete characterization of when $ζ_A(z)$ is algebraic and prove that it admits the circle of convergence as a natural boundary in the transcendence case. This is in stark contrast to the case of linear endomorphisms on $\mathbb{R}^d/\mathbb{Z}^d$ in which Baake, Lau, and Paskunas prove that the zeta function is always rational. Some connections to earlier work of Bell, Byszewski, Cornelissen, Miles, Royals, and Ward are discussed. Our method uses a similar technique in recent work of Bell, Nguyen, and Zannier together with certain patching arguments involving linear recurrence sequences.
