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Let $Q\in{\mathbb Z}[X], Q=\sum_{k=0}^m q_kX^k$, with degree $m$ and Galois group $S_m$ (over $\mathbb Q$). Consider the evenized-reciprocalized polynomial $P(X)=X^{2m}Q\bigg(X^2+\frac{1}{X^2}\bigg)$.

Is it true that, except on a set of Zariski dimension $0$ (I mean, in terms of the coefficients $q_0,q_1,\ldots, q_m$), the Galois group of $P$ (over $\mathbb Q$) is $V_4 \wr S_m$ ?

This question is natural in the context of another recent question of mine.

Jyrki Lahtonen
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Ewan Delanoy
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  • "Zariski dimension" (the dimension of the Zariski closure I guess?) is not really the right measure here. After all, the Galois group of $X^2 - t$ as $t$ varies is $S_2$ for all $t$ which are not squares, but rational squares are Zariski dense in $\mathbf{A}^1$. The right concept is a "thin" set. Actually, by Hilbert irreducibility, this is immediate once you verify the Galois group is correct over the function field $\mathbf{Q}(q_0,\ldots,q_m)$. – user760870 May 20 '20 at 19:06
  • But the function field case is clear. If $L = \mathbf{Q}(x_1,\ldots,x_n)$, and $K = \mathbf{Q}(q_1,\ldots,q_n)$ where $s_i$ are the symmetric polynomials in $x_i$, and $M = \mathbf{Q}(y_1,\ldots,y_n)$ where $y^2_n + y^{-2}_n = x_n$ so $(y_n + y^{-1}_n)^2 = x_n + 2$, then $\mathrm{Gal}(M/K) = V_4 \wr S_n$ almost immediately. – user760870 May 20 '20 at 19:12
  • @user760870 Do consider fleshing that out to an answer! While you are at it, could you recommend me a source to study Hilbert irreducibility, specializations and such. I have somehow managed to never come in touch with the theme. Of course, it may turn out that I'm no longer man enough to learn it, but that's a future problem. If you prefer, I can make that a separate question. – Jyrki Lahtonen May 20 '20 at 19:56
  • Dear @JyrkiLahtonen, one could probably do worse than read these notes from a course by Serre (http://www.msc.uky.edu/sohum/ma561/notes/workspace/books/serre_galois_theory.pdf). The example of $t \in \mathbf{A}^1$ and the family $x^2 - t$ is very much the one to keep in mind --- you have to avoid the "thin set" consisting of the image of the rational points of a degree $\ge 2$ map, in this case $\mathbf{A}^1 \rightarrow \mathbf{A}^1$ given by $z \mapsto z^2$. One then wants to show that if you count (by hight) points in the base, the thin sets contribute $o(1)$ to the count. – user760870 May 21 '20 at 00:29
  • Thanks @user760870 – Jyrki Lahtonen May 21 '20 at 06:52
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    @user760870 I'm struggling to understand and restate in more elementary terms the part from Serre's book that helps with my current problem. Hence this question : (see next comments) – Ewan Delanoy May 21 '20 at 07:02
  • @user760870 Say that a set $A \subseteq {\mathbb Q}^d$ is "thin in Ewan's sense" if it is a finite union of elementray thin sets, and an elementary thin set is either a set of tuples that satisfy a polynomial equation (involving at least one variable to avoid trivialties), or a set of tuples where one coordinate is a rational function in the other coordinates plus one extra parameter (and the rational function must be non-constant in at least one of the other coordinates, to avoid the triviality of defining the distinguished coordinate as simply equal to the parameter). – Ewan Delanoy May 21 '20 at 07:03
  • @user760870 Then, does it follow from the results in Serre's book that the set of $(a_0,\ldots,a_d)$ for which $X^d+a_{d-1}X^{d-1}+\ldots+a_1X+a_0$ is not irreducible or does not have $S_n$ as a Galois group, is thin in Ewan's sense ? – Ewan Delanoy May 21 '20 at 07:04
  • Another reference: Ch 9 of Lectures on the Mordell-Weil Theorem also by Serre (https://epdf.pub/queue/lectures-on-the-mordell-weil-theorem.html). @JyrkiLahtonen. – user760870 May 21 '20 at 21:27
  • @EwanDelanoy I can't even parse "rational function in the other coordinates plus one extra parameter" but maybe look at the second reference I gave above. – user760870 May 21 '20 at 21:28
  • @user760870 That second document is only provided in some strange format (may be e-book or something). My system cannot read it. – Jyrki Lahtonen May 22 '20 at 04:39
  • @JyrkiLahtonen I had the same problem with the second document, but I solved it by converting djvu to pdf, using the online tool at https://djvu2pdf.com – Ewan Delanoy May 22 '20 at 07:36
  • @user760870 I'm curious if you know the answer to https://math.stackexchange.com/questions/3709277/question-about-zariski-density-and-polynomials-with-full-galois-group – Ewan Delanoy Jun 07 '20 at 08:38

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