Arkiv för Matematik

Volume 55 (2017)

Number 1

Algebraic independence of the values of power series with unbounded coefficients

Pages: 61 – 87

DOI: https://dx.doi.org/10.4310/ARKIV.2017.v55.n1.a3

Author

Kaneko Hajime (Institute of Mathematics, University of Tsukuba, Ibaraki, Japan; and Center for Integrated Research in Fundamental Science and Engineering (CiRfSE), University of Tsukuba, Ibaraki, Japan)

Abstract

Many mathematicians have studied the algebraic independence over $\mathbb{Q}$ of the values of gap series, and the values of lacunary series satisfying functional equations of Mahler type. In this paper, we give a new criterion for the algebraic independence over $\mathbb{Q}$ of the values $\sum^{\infty}_{n=0} t(n) \beta^{-n}$ for distinct sequences $(t(n))^{\infty}_{n=0}$ of nonnegative integers, where $\beta$ is a fixed Pisot or Salem number. Our criterion is applicable to certain power series which are not lacunary. Moreover, our criterion does not use functional equations. Consequently, we deduce the algebraic independence of certain values $\sum^{\infty}_{n=0} t_1 (n) \beta^{-n} , \dotsc , \sum^{\infty}_{n=0} t_r( n) \beta^{-n}$ satisfying\[\lim_{n \to \infty , t{i-1} (n) \neq 0} \; \dfrac{t_i(n)}{t_{i-1}(n)^M} = \infty \; (i=2, \dotsc, r)\]for any positive real number $M$.

Keywords

algebraic independence, Pisot numbers, Salem numbers

2010 Mathematics Subject Classification

Primary 11J99. Secondary 11K16, 11K60.

Received 1 November 2016

Received revised 12 March 2017

Accepted 9 April 2017

Published 26 September 2017