Hypertranscendence of solutions of Mahler equations
Hypertranscendence of solutions of Mahler equations
复制标题
马勒方程解的超超越
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
J. Roques
中科院分区：
文献类型：
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作者：
T. Dreyfus;C. Hardouin;J. Roques
在过去的几年里，数学家们对马勒函数的兴趣越来越大。这类功能包括自动序列的生成系列。本文研究了组合数学中普遍存在的下列问题：一组Mahler函数$u_{1}，.给定u_{n}$，则为$u_{1}，.，u_{n}$及其连续导数代数无关？在本文中，我们给出了一般标准，确保肯定的答案，这个问题。我们将我们的主要结果生成系列附加到所谓的鲍姆甜和Rudin-Shapiro自动序列。特别是，我们表明，这些系列是超代数独立的，即，这些级数和它们的连续导数是代数独立的。我们的方法依赖于参数化的差分伽罗瓦理论（在这种情况下，一个给定的马勒方程的解决方案之间的代数微分关系反映了一个线性微分代数群）。
The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is omnipresent in combinatorics: a set of Mahler functions $u_{1},...,u_{n}$ being given, are $u_{1},...,u_{n}$ and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalgebraically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).