On the Eisenstein constant (Q936469)

Eisenstein proved that if an entire series \(u=\sum_{n\geq0} \alpha_nz^n\) with rational coefficients represents an algebraic function, then there exists an integer \(a\) (called an Eisenstein constant) such that \(a^n\alpha_n\) is an integer for all \(n\). Precise values for Eisenstein constants, or upper bounds for them, are known in a relatively small number of cases [see for instance \textit{W. M. Schmidt}, ``Eisenstein's theorem on power series expansions of algebraic functions'', Acta Arith. 56, 161--179 (1990; Zbl 0659.12003) and \textit{B. M. Dwork} and \textit{A. J. van der Poorten}, ``The Eisenstein constant'', Duke Math. J. 65, 23--43 (1992; Zbl 0770.11051)]. The main result of the paper is a criterion which allows to produce Eisentein constants for regular algebraic functions. The method is explained in detail for functions of type \((1-x)^c\), where \(c\) is a positive rational mumber.