TY - JOUR
AU - Baker, A.
AB - A. BAKER A classical theorem of Eisensteinf states that if the series a.(x) = a +a x+a x +... (1) o 1 2 has rational coefficients and represents an algebraic function, regular at the origin, then there is an integer q such that q a is an integer for each n > 0. In the present note we show that there is a wide class of algebraic functions for which this theorem is best possible, in the sense that an n n x integer q exists such that, for all n, q a is an integer and q ~ a is not. n n An algebraic function w = a (a;) is the root of an equation of the form P (x)w»+P _ (x)w*- +... + P (x)w+P (x) = 0, (2) N N 1 1 Q where N is a positive integer and P {x), ..., P (x) are polynomials in x. Q N We shall consider equations of the form (2) for which the following two conditions are satisfied. (i) The constant coefficients in the polynomials P^x) are 1 for j = 0, 1 and 0forj=r-2, 3, ...,N. (ii) All the other coefficients are integers except for 
TI - Power Series Representing Algebraic Functions
JF - Journal of the London Mathematical Society
DO - 10.1112/jlms/s1-40.1.108
DA - 1965-01-01
UR - https://www.deepdyve.com/lp/wiley/power-series-representing-algebraic-functions-HTB6J6P885
SP - 108
EP - 110
VL - s1-40
IS - 1
DP - DeepDyve
ER -