AI 中文总结
研究多元马勒函数的函数性质，证明其在特定区域的解析性与亚纯性及有理 - 超越二分法，加强相关提升定理，能计算函数值间代数关系，还证明了正则点情况下的多元下降定理。
AI 中文摘要
我们研究了与一大类马勒变换相关的多元马勒函数的函数性质。证明了这些函数在零点邻域总是解析的，在\(C^n\)中的开单位球以及对所有素数\(p\)在\(C_n^p\)上是亚纯的。这使我们能够证明这些函数的有理 - 超越二分法。结果，加强了Adamczewski和Faverjon最近得到的提升定理。该定理能在已知函数自身代数关系的情况下，系统地计算由马勒系统相关的多元马勒函数值之间的代数关系。最后，我们在正则点的情况下证明了一个多元下降定理。
英文摘要
We study the functional properties of Mahler functions of several variables associated with a large class of Mahler transformations. We prove that these functions are always analytic on a neighborhood of zero, and meromorphic on the open unit ball in C^n , as well as in C_n^p for all primes p. This enables us to prove a rational-transcendental dichotomy for these functions. As a consequence, we strengthen the lifting theorem recently obtained by Adamczewski and Faverjon. This theorem allows us to systematically compute the algebraic relations between values of multivariate Mahler functions related by a Mahler system, given the algebraic relations between the functions themselves. Finally we prove a multivariate descent theorem in the case of regular points.
