## Month: April, 2018

### 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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### Minimal and maximal matrix convex sets

The final version of the paper Minimal and maximal matrix convex sets, written by Ben Passer, Baruch Solel and myself, has recently appeared online. The publisher (Elsevier) sent us a link through which the official final version is downloadable, for anyone who clicks on the following link before May 26, 2018. Here is the link for the use of the public:

Abstract. For every convex body $K \subseteq \mathbb{R}^d$, there is a minimal matrix convex set $\mathcal{W}^{min}(K)$, and a maximal matrix convex set $\mathcal{W}^{max}(K)$, which have $K$ as their ground level. We aim to find the optimal constant $\theta(K)$ such that $\mathcal{W}^{max}(K) \subseteq \theta(K) \cdot \mathcal{W}^{min}(K)$. For example, if $\overline{\mathbb{B}}_{p,d}$ is the unit ball in $\mathbb{R}^d$ with the $p$-norm, then we find that
$\theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p-1/2|}$ .
This constant is sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which $\mathcal{W}^{max}(K) \subseteq \mathcal{W}^{min}(L)$. In particular, we obtain that a convex body $K$ satisfies $\mathcal{W}^{max}(K) = \mathcal{W}^{min}(K)$ only if $K$ is a simplex.
These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every $d$-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit constructions of these (and other) dilations.