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 $z=0$ .

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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Linear algebra over rings

Test your intuition: is the following true or false?

Assertion 1: If $A$ is a square matrix over a commutative ring $R$ , the rows of $A$ are linearly independent over $R$ if and only if the columns of $A$ are linearly independent over $R$ .

(All rings in this post will be nonzero commutative rings with identity.)

And how about the following generalization?

Assertion 2: If $A$ is an $m \times n$ matrix over a commutative ring $R$ , the row rank of $A$ (the maximum number of $R$ -linearly independent rows) equals the column rank of $A$ (the maximum number of $R$ -linearly independent columns).

If you want to know the answers, read on…

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Fitting ideals of modules

In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I’d like to explain how the uniqueness portion of that theorem is actually a special case of a more general result, called Fitting’s Lemma, which holds for arbitrary commutative rings.

We begin by proving that one can characterize the diagonal entries in the Smith Normal Form of a matrix $A$ over a PID in an intrinsic way by relating them to the GCD of the $k \times k$ minors of $A$ for all $k$ . Actually, since the GCD isn’t defined for general rings, we will instead consider the ideal generated by the $k \times k$ minors (which makes sense for any ring, and is the ideal generated by the GCD in the case of a PID).

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Finitely generated modules over a P.I.D. and the Smith Normal Form

I’m teaching Graduate Algebra this semester, and I wanted to record here the proof I gave in class of the (existence part of the) structure theorem for finitely generated modules over a PID. It’s a standard argument, based on the existence of the Smith Normal Form for a matrix with entries in a PID, but it’s surprisingly hard to find a concise and accessible reference.

We assume familiarity with basic definitions in the theory of modules over a (commutative) ring. Our goal is to prove the following:

Theorem: Let $R$ be a principal ideal domain and let $M$ be a finitely generated $R$ -module. Then $M$ is isomorphic to a direct sum of cyclic $R$ -modules. More precisely, there are a non-negative integer $r$ (called the rank of $M$ ) and elements $d_1,\ldots,d_n \in M$ (called the invariant factors of $M$ ) such that $d_i \mid d_{i+1}$ for all $i=1,\ldots,n-1$ and $M \cong R^r \oplus R/(d_1) \oplus R/(d_2) \oplus \cdots \oplus R/(d_n)$ .