Algebraic Values of Transcendental Functions at Algebraic Points

In honor of Pi Day 2023, I’d like to discuss Hilbert’s 7th Problem, which in an oversimplified (and rather vague) form asks: under what circumstances can a transcendental function take algebraic values at algebraic points?

The connection with \pi is that Lindemann proved in 1882 that the transcendental function f(z) = e^z takes transcendental values at every nonzero algebraic number. Since e^{\pi i} = -1 by Euler’s formula, this proves that \pi i, and hence \pi itself, is transcendental. In light of this theorem, it is natural to wonder what if anything is special here about the function f(z) = e^z and the point z=0.

Ferdinand von Lindemann

One thing that’s special about z=0 is that if \alpha \neq 0 is algebraic and e^\alpha is also algebraic, then both n\alpha and e^{n \alpha} are algebraic for all n \in {\mathbb Z}, and these numbers are all distinct. So one might be led to speculate that if f is a transcendental entire function then there are only finitely many algebraic numbers \alpha for which f(\alpha) is also algebraic.

Unfortunately, as Hilbert knew, this is completely false. For example, the function f(z) = e^{2\pi iz} is transcendental but it takes the rational value 1 at every integer. In 1886, Weierstrass had given an example of a transcendental entire function that takes rational values at all rational numbers; later, in 1895, Stäckel showed that there is a transcendental entire function that takes rational values at all algebraic points. However, the functions of Weierstrass and Stäckel, are in some sense “pathological”; they have large growth rates and do not occur “in nature”. The challenge is to make this intuitive feeling more precise, and also to distinguish e^z from e^{2\pi iz}.

One thing that is special about e^z, which is not shared by any of the other functions mentioned in the previous paragraph, is that it satisfies a linear differential equation with rational coefficients (namely f'(z) = f(z)). The existence of such a (not necessarily linear) differential equation turns out to be the key idea needed to generalize Lindemann’s theorem in a substantial way.

Another fruitful generalization is to rephrase our original question as an unlikely intersection problem: given two algebraically independent entire functions f_1(z) and f_2(z) satisfying suitable hypotheses, can we conclude that there are only finitely many complex numbers \alpha such that f_1(\alpha) and f_2(\alpha) are simultaneously algebraic? This generalizes our original question by letting f_1(z) = z and f_2(z) = f(z).

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