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Fix natural numbers $d,N$ and a polynomial $\Delta \in \mathbb{C}[x_1,\ldots,x_d]$. Let $S_{d,N}$ be the set of field extensions $K/ \mathbb{C}(x_1,\ldots,x_d)$ such that

  1. The degree $[K: \mathbb{C}(x_1,\ldots,x_d)]$ is bounded by $N$.
  2. The discriminant of $K/ \mathbb{C}(x_1,\ldots,x_d)$ is $\Delta$.
  3. $K$ is generated by an element whose minimal polynomial has coefficients in $\mathbb{C}[x_1,\ldots,x_d]$ that have degrees at most $N$.

Question: Is $S_{d,N}$ finite?

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1 Answer 1

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Yes, and I don't think you need 3. Let $D\subset \mathbb{C}^d$ be the locus where $\Delta $ vanishes. You are looking at étale covers of a fixed degree $n\ (\leq N)$ of $\mathbb{C}^d\smallsetminus D$, up to birational isomorphism. Such covers are classified by an action of $\pi _1(\mathbb{C}^d\smallsetminus D)$ on a set with $n$ elements. Since $\pi _1(\mathbb{C}^d\smallsetminus D)$ is finitely generated there are only finitely many possibilities.

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  • $\begingroup$ Thank you very much for your answer. We still do not understand one thing: why is the field extension $K/ \mathbb{C}(x_1,\ldots,x_d)$ uniquely determined by the action of $\pi _1(\mathbb{C}^d\smallsetminus D)$ on a set with $n$ elements (in other words, why is the extension uniquely determined by the isomorphism class of the corresponding cover of $\mathbb{C}^d\smallsetminus D$ in the analytic category). Does it follow from some compactification argument or some relative version of GAGA? If so, can you point us to a relevant reference? Thank you very much again. $\endgroup$
    – Rami
    Commented Aug 1, 2023 at 15:13
  • $\begingroup$ The cover $X$ of $\mathbb{C}^d\smallsetminus D$ is algebraic, so $K$ is just its field of rational functions, and the extension corresponds to the covering $X\rightarrow \mathbb{C}^d\smallsetminus D$. There is no need for compactification or GAGA argument, just the theory of the algebraic fundamental group. $\endgroup$
    – abx
    Commented Aug 1, 2023 at 15:56
  • $\begingroup$ Thank you very much. We write here the relevant references for the record: 1. [SGA1, Expos´e XII, Th´eor`eme 5.1]. library.msri.org/books/sga/sga/1/1t_332.html 2. stacks.math.columbia.edu/tag/0BND 3. people.math.ethz.ch/~pink/Theses/2018-Bachelor-Noah-Held.pdf Theorem 6.9. $\endgroup$
    – Rami
    Commented Aug 6, 2023 at 10:16

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