The ring of entire functions

Rings made a bad first impression on me. I couldn’t remember the definitions of all the different kinds of rings, much less have an intuition for what was important about each one. As I recall, all the examples of rings in our course were variations on the integers, often artificial variations.

Entire functions

I’m more interested in analysis than algebra, so my curiosity was piqued when I ran across an appendix on entire functions in the back of an algebra book [1]. This appendix opens with the statement

The ring E of entire functions is a most peculiar ring.

It’s interesting that something so natural from the perspective of analysis is considered peculiar from the perspective of algebra.

An entire function is a function of a complex variable that is analytic in the entire complex plane. Entire functions are functions that have a Taylor series with infinite radius of convergence.

Rings

A ring is a set of things with addition and multiplication operations, and these operations interact as you’d expect via distributive rules. You can add, subtract, and multiply, but not divide: addition is invertible but multiplication is not in general. Clearly the sum or product of entire functions is an entire function. But the reciprocal of an entire function is not an entire function because the reciprocal has poles where the original function has zeros.

So why is the ring of analytic functions peculiar to an algebraist? Osborne speaks of “the Jekyll-Hyde nature of E,” meaning that E is easy to work with in some ways but not in others. If Santa were an algebraist, he would say E is both naughty and nice.

Nice properties

On the nice side, E is an integral domain. That is, if f and g are entire functions and fg = 0, then either f = 0 or g = 0.

If we were looking at functions that were merely continuous, it would be possible for f to be zero in some places and g to be zero in the rest, so that the product fg is zero everywhere.

But analytic functions are much less flexible than continuous functions. If an analytic function is zero on a set of points with a limit point, it’s zero everywhere. If every point in the complex plane is either a zero of f or a zero of g, one of these sets of zeros must have a limit point.

Another nice property of E is that it is a Bezóut domain. This means that if f and g are entire functions with no shared zeros, there exist entire functions λ and μ such that

λf + μg = 1.

This is definition is analogous to (and motivated by) Bezóut’s theorem in number theory which says that if a and b are relatively prime integers, then there are integers m and n such that

ma + nb = 1.

Naughty properties

The naughty properties of E take longer to describe and involve dimension. “Nice” rings have small Krull dimension. For example Artinian rings have Krull dimension 0, and the ring of polynomials in n complex variables has dimension n. But the Krull dimension of the ring of entire functions is infinite. In fact it’s “very infinite” in the sense that it is at least

$2^{\aleph_1}$

and so the Krull dimension of E is larger than the cardinality of the complex numbers.

Wittgenstein’s ruler

Nassim Taleb described Wittgenstein’s ruler this way:

Unless you have confidence in the ruler’s reliability, if you use a ruler to measure a table you may also be using the table to measure the ruler. The less you trust the ruler’s reliability, the more information you are getting about the ruler and the less about the table.

An algebraist would say that entire functions are weird, but an analyst could say that on the contrary, ring theory, or at least Krull dimension, is weird.

[1] Basic Homological Algebra by M. Scott Osborne.