The ring of entire functions

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Rings made a bad first impression on me. I could not remember the definitions of all the different types of rings, much less I have an intuition for what is important in each. Remember, all the examples of rings in our course were variations on integers, mostly artificial variations.

Complete functions

I’m more interested in analysis than algebra, so my curiosity arose when I came across an appendix on whole functions at the back of an algebra book. [1]. This appendix opens with a statement

The ring God Of whole functions is an extremely strange ring.

It is interesting that something so natural from the point of view of analysis is considered strange from the point of view of algebra.

A The whole function Is a function of a complex variable that is analytical in the whole complex plane. Complete functions are functions that have a Taylor series with an infinite convergence radius.


A ring is a set of things with addition and multiplication operations, and these operations interact as you would expect through distributive laws. You can add, subtract and multiply, but not divide: the connection is reversible but the multiplication is not in general. It is clear that the sum or product of whole functions is a whole function. But the reciprocity of a whole function is not a whole function because reciprocity has poles where the original function has zeros.

So why is the ring of analytical functions unique to Algebraist? Osborne speaks of the “Jekyll-Hyde nature of God,” That is God It is easy to work with in some ways but not in others. If Santa was an algebraist, he would say God He is both naughty and nice.

Nice properties

On the nice side, God he Integral domain. That is, if and and P Are complete functions and fg = 0, so or and = 0 or P = 0.

If we were to look at functions that were just continuous, it would have been possible and Be zero in some places and P Be zero in the rest, so the product fg He’s zero everywhere.

But analytical functions are much less flexible than continuous functions. If an analytical function is zero on a set of points with a boundary point, it is zero everywhere. If each point on the complex plane is zero of and Or zero of P, One of these zero groups must have a boundary point.

Another nice property of God Is that it a Bezóut domain. It means that if and and P Are whole functions without common zeros, there are whole functions λ and μ such

λand + μP = 1.

This definition is equivalent to (and driven by) a trivial statement in number theory that says if A and B They are relatively prime integers, so there are integers M and N so that

ma + Note: = 1.

Naughty assets

The playful characteristics of God It takes longer to describe and involve a dimension. “Nice” rings have small ones Carol dimension. For example, earth rings have a Carol 0 dimension, and the polynomial ring b N Complex variables have a dimension N. But the caroll dimension of the ring of complete functions is infinite. In fact it is “very infinite” in the sense that at least

where A1 Greater than ℵ0, The next cardinal after ℵ0 If you get the sequence hypothesis. So the carol dimension of God Greater than the cardinality of the composite numbers.

The ruler of Wittgenstein

Nassim Taleb described Shalit Wittgenstein as follows:

Unless you have confidence in the reliability of the bar, if you use the bar to measure a table you may also use the table to measure the bar. The less you trust the reliability of the toolbar, the more information you get about the toolbar and less about the table.

Algebraist would say that whole functions are strange, but an analyst can say that on the contrary, the theory of rings, or at least the Carol dimension, is strange.

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[1] Basic homologous algebra By M. Scott Osborne.




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