Noncommutative Analysis

Category: Fun stuff

The disc trick (and some other cute moves)

This post is about a chain of little tricks that I discovered with collaborators and used in several papers. It is just a collection of simple moves that lets you deduce the existence of a zero preserving map of a certain class between two gauge invariant spaces, given the existence of a map from that class (things will be very clear soon, I hope). These tricks were later used by some other people, who applied it in different settings.

I am writing this post as notes for my upcoming Pizza & Beer seminar talk. The section at the end of the notes contains references and links to papers where this was used.

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The perfect Nullstellensatz just got more perfect

After giving a talk about the perfect Nullstellensatz (the commutative free Nullstellensatz) at the Technion Math department’s pizza and beer seminar, I had a revelation: I think it holds over other fields as well, not just over the complex numbers! (And in particular, contrary to what I thought before, it holds over the reals. It seems to hold over other fields as well). 

To explain, I will need some notation. 

Let k be a field. We write A = k[z_, \ldots, z_d] – the algebra of all polynomials in d (commuting) variables over the field k

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Polya’s three rules of style

In G. Polya‘s book “How to Solve It”, one of the shortest sections is called “Rules of style”. This section contains Polya’s three rules of style, which are worth repeating.

“The first rule of style”, writes Polya, “is to have something to say”.

“The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time”.

Polya’s third rule of style is: “Don’t say what does not need to be said” or maybe “Don’t say the obvious”. I am not sure of the exact formulation, because Polya doesn’t write the third rule down – that would be a violation of the rule!

Polya’s three rules are excellent and one is advised to follow them if one strives for good style when writing mathematics. However, style is not the only criterion by which we measure mathematical writing. There is a tradeoff between succinct and elegant style, on the one hand, and clarity and precision, on the other.

“Don’t say the obvious” – sure! But what is obvious? And to whom? A careful writer leaving a well placed exercise in a textbook is one thing. An author of a long and technical paper that leaves an exercise to the poor, overworked referee, is something different. And, of course, a mathematician leaving cryptic notes to his four-months-older self, is the most annoying of them all.

“Don’t say the obvious” – sure, sure! But is it even true? I think that all the mistakes that I am responsible for publishing have originated by an omission of an “obvious” argument. I won’t speak about actual mistakes made by others, but I do have the feeling that some people have gotten away with not explaining something non-trivial, and were lucky that things turned out to be as their intuition suggested (granted, having the correct intuition is also a non-trivial achievement).

I disagree with Polya’s third rule of style. And you see, to reject it, I had to formulate it. QED.

The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field \mathbb{C} (e.g., there are many polynomials, such as x^{2n} + 1, that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write \mathbb{C}[z] for the algebra of polynomials in one complex variable with complex coefficients, and consider a polynomial as a function of the complex variable z \in \mathbb{C}. We will also write \mathbb{C}[z_1, \ldots, z_d] for the algebra of polynomials in d (commuting) variables, and think of polynomials in \mathbb{C}[z_1, \ldots, z_d] – at least initially – as a functions of the variable z = (z_1, \ldots, z_d) \in \mathbb{C}^d

[Update June 24, 2019: contrary to what I thought, the main theorem presented below holds over arbitrary fields, not just over the complex numbers, very much by the same proof. See this post.]

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Journal of Xenomathematics

I am happy to advertise the existence of a new electronic journal/forum/website: Journal of Xenomathematics. Don’t worry, it’s not another new research journal. The editor is John E. McCarthy. The purpose is to discuss mathematics that is out of this world. Aren’t you curious?