Category: Algebraic geometry

Around and under my talk at Fields

This week I am attending a Workshop on Developments and Technical Aspects of Free Noncommutative Functions at the Fields Institute in Toronto. Since I plan to give a chalk-talk, I cannot post my slides online (and I cannot prepare for my talk by preparing slides), so I will write here what some ideas around what I want to say in my talk, and also some ramblings I won’t have time to say in my talk.

[Several years ago I went to a conference in China and came back with the insight that in international conferences I should give a computer presentation and not a blackboard talk, because then people who cannot understand my accent can at least read the slides. It’s been almost six years since then and indeed I gave only beamer-talks since. My English has not improved over this period, I think, but I have several reasons for allowing myself to give an old fashioned lecture – the main ones are the nature of the workshop, the nature of the audience and the kind of things I have to say].

In the workshop Guy Salomon, Eli Shamovich and I will give a series of talks on our two papers (one and two). These two papers have a lot of small auxiliary results, which in usual conference talk we don’t get the chance to speak about. This workshop is a wonderful opportunity for us to highlight some of these results and the ideas behind them, which we feel might be somewhat buried in our paper and have gone unnoticed.

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 $z = (z_1, \ldots, z_d) \in \mathbb{C}^d$

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