### 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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