On vanishing coefficients of algebraic power series over fields of positive characteristic
On vanishing coefficients of algebraic power series over fields of positive characteristic
复制标题
关于正特征域上代数幂级数的消失系数
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
J. Bell
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文献类型：
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作者：
B. Adamczewski;J. Bell
设K是一个特征p>0的域，f（t1，...，td）是多元有理函数域K（t1，...，td）上的一个系数为k的d元幂级数.我们证明了Derksen最近的Skolem-Mahler-Lech定理的正特征类似定理和Christol的一个经典定理的一个推广，证明了使得f（t1，...，td）中的$t_{1}^{n_{1}}cdots t_{d}^{n_{d}}$的系数为零的指数集（n1，...，nd）是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-Lang定理领域的积极特征。
Let K be a field of characteristic p>0 and let f(t1,…,td) be a power series in d variables with coefficients in K that is algebraic over the field of multivariate rational functions K(t1,…,td). 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 (n1,…,nd)∈ℕd for which the coefficient of $t_{1}^{n_{1}}cdots t_{d}^{n_{d}}$ in f(t1,…,td) 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.