A necessary and sufficient condition for transcendency

Abstract:

As has been known for many years (see, e.g., K. Mahler, J. Reine Angew. Math., v. 166, 1932, pp. 118-150), a real or complex number $\zeta$ is transcendental if and only if the following condition is satisfied. To every positive number $\omega$ there exists a positive integer n and an infinite sequence of distinct polynomials $\{ {p_r}(z)\} = \{ {p_{{r_0}}} + {p_{{r_1}}}z + \cdots + {p_{{r_n}}}{z^n}\}$ at most of degree n with integral coefficients, such that \[ 0 < |{p_r}(\zeta )| \leqslant {\{ p_{{r_0}}^2 + p_{{r_1}}^2 + \cdots + p_{{r_n}}^2\} ^{ - \omega }}\quad for\;all\quad r.\] In the present note I prove a simpler test which makes the transcendency of $\zeta$ depend on the approximation behaviour of a single sequence of distinct polynomials of arbitrary degrees with integral coefficients.