A Necessary and Sufficient Condition for Transcendency

As has been known for many years (see, e.g., K. Mahler, /. Reine Angew. Math., v. 166, 1932, pp. 118-150), a real or complex number f is transcendental if and only if the following condition is satisfied. To every positive number co there exists a positive integer n and an infinite sequence of distinct polynomials {p,(z)} = {pr + pr z + • • • + p z } at most of 0 1 n degree n with integral coefficients, such that 0 < \pr(i)l < {p2o + p2x + ■ ■ ■ + P2Y" for all r. In the present note I prove a simpler test which makes the transcendency of f depend on the approximation behaviour of a single sequence of distinct polynomials of arbitrary degrees with integral coefficients. 1. If n n P(z) = Z PhzH =Pn i! (Z ~ aft)> where Pn * °> h = 0 h=l is any polynomial with real or complex coefficients, of the exact degree «, and with the zeros a,, ■ • ■ , an, put dip) = n, Mip) = exp ( f log |p(e27r/í)|dí), m(p) = +V ¿ \ph \ I h = 0 It is well known that (1) Mip) = \pn | rj max (1, \ah I), Mip) < mip). h=l Next, if f is any real or complex number, put (1 if Ms real, (2 otherwise, and denote by s$(f) the set of all polynomials p(z) with integral coefficients that satisfy the inequality p(f) ¥= 0. Received January 23, 1974. AMS (MOS) subject classifications (1970). Primary 10F35. Copyright © 1975, American Mathematical Society 145 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use