Is there a reasonable definition of the height of a transcendental number - MathOverflow

Is there a reasonable definition of the height of a transcendental number

2011-10-12T08:06:38Z

For an algebraic number $\alpha$ one can define its "height" in many ways. Informally, you could use its minimal polynomial over $\mathbf{Q}$ and consider the maximum of the heights of its coefficients. Or consider all the valuations of $\alpha$, etc. In this context, the height is supposed to be some kind of measure of complexity.

Question. Is there a reasonable definition of the "height" of a transcendental number.

I'm not sure what such a height would mean though in this context.

If there isn't any reasonable definition, is there any reasonable explanation for why this isn't possible?

Answer by S. Carnahan for Is there a reasonable definition of the height of a transcendental number

2011-10-13T19:47:54Z

Here is one potentially reasonable explanation for why such an invariant shouldn't exist. One property of height that can be useful is that the height of an algebraic number is invariant under all automorphisms of all rings that contain that number. For any algebraically independent pair of transcendental complex numbers, one may choose a ring-theoretic automorphism of $\mathbb{C}$ that exchanges them. If we want the same sort of invariance as in the algebraic setting, then all heights of transcendental numbers must be equal.

I think the existence of a single correct answer to your question would require a specific application.

Answer by Adrien for Is there a reasonable definition of the height of a transcendental number

2011-10-13T19:58:11Z

It does'nt quite answer your question, but maybe there is a resaonable definition of the height of a period. The ring of period is a countable over-ring of the field of algebraic numbers, sharing (at least conjecturally) many properties with them. This ring contains most of interesting transcendental numbers.