A necessary and sufficient condition for transcendency
HTML articles powered by AMS MathViewer
- by K. Mahler PDF
- Math. Comp. 29 (1975), 145-153 Request permission
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.References
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Math. Comp. 29 (1975), 145-153
- MSC: Primary 10F35
- DOI: https://doi.org/10.1090/S0025-5718-1975-0382184-0
- MathSciNet review: 0382184