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arXiv 1205.4091math.NT
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关于正特征域上代数幂级数的消失系数

On vanishing coefficients of algebraic power series over fields of positive characteristic

Boris Adamczewski, Jason P. Bell

发表信息：
Inventiones Mathematicae 187 (2012) 343--393
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摘要：

设$K$是特征$p>0$的域，$f(t_1,...,t_d)$是多元有理函数域$K(t_1,...,t_d)$上的$K$中系数为$d$变量的幂级数.我们证明了一个推广的Derksen最近模拟的Skolem-Mahler-Lech定理的正特征和一个经典的Christol定理，通过显示的指数集$(n_1,...,n_d)\in \mathbb{N}^d$的系数$t_1^{n_1}...t_d^{n_d}$在$f(t_1,...,t_d)$是零是一个$p$ -自动集。应用这一结果多元有理函数导致有趣的有效结果有关的一些丢番图方程有关的$S$单位方程和更普遍的Mordell-郎定理领域的积极特征。

英文摘要：

Let $K$ be a field of characteristic $p>0$ and let $f(t_1,...,t_d)$ be a power series in $d$ variables with coefficients in $K$ that is algebraic over the field of multivariate rational functions $K(t_1,...,t_d)$. We prove a generalization of both Derksen's recent analogue of the Skolem-Mahler-Lech theorem in positive characteristic and a classical theorem of Christol, by showing that the set of indices $(n_1,...,n_d)\in \mathbb{N}^d$ for which the coefficient of $t_1^{n_1}...t_d^{n_d}$ in $f(t_1,...,t_d)$ is zero is a $p$-automatic set. Applying this result to multivariate rational functions leads to interesting effective results concerning some Diophantine equations related to $S$-unit equations and more generally to the Mordell--Lang Theorem over fields of positive characteristic.

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