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Mathematics of Computation

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A necessary and sufficient condition for transcendency

Author: K. Mahler
Journal: Math. Comp. 29 (1975), 145-153
MSC: Primary 10F35
MathSciNet review: 0382184
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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

$\displaystyle 0 < \vert{p_r}(\zeta )\vert \leqslant {\{ p_{{r_0}}^2 + p_{{r_1}}^2 + \cdots + p_{{r_n}}^2\} ^{ - \omega }}\quad for\;all\quad r.$

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.

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Article copyright: © Copyright 1975 American Mathematical Society