Бесконечная линейная и алгебраическая независимость значений -рядов в полиадических лиувиллевых точках

Abstract

This paper proves infinite linear and algebraic independence of the values of 𝐹-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. 𝐹- series have form 𝑓𝑛 =Σ∞𝑛=0 𝑎𝑛𝑛!𝑧𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties. These series converge in the field Q𝑝 of 𝑝-adic numbers and their algebraic extensions K𝑣. Polyadic number is a series of the form Σ∞𝑛=0 𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞|< 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼−𝐴|𝑝 < 𝐴−𝐷. Infinite linear (algebraic) independence means that for any nonzero linear form (any nonzero polynomial) there are infinitely many primes 𝑝 and valuations 𝑣 extending 𝑝-adic valuation to an algebraic number field K with the following property: the result of substitution in the considered linear form (polynomial) of the values of 𝐹 - of series instead of variables is a nonzero element of the field. Previously, only the existence of at least one prime number 𝑝 with the properties listed above was proved.