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Let $G=\mathrm{SO}_{2 n}$ (or $G=\mathrm{SO}_{2n+1}$, $G=\mathrm{Sp}_{2 n}$ …) defined over some field $K$. Consider $G$ as an affine subvariety of the space of matrices.

  1. (Warm-up question) What are the asymptotics of $\deg(G)$? There's Brandt, Bruce, Brysiewicz, Krone, and Robeva - The degree of $\operatorname{SO}(n)$ (surprisingly recent!), and one should be able to get asymptotics from that, but that's not obvious to someone who isn't really a combinatorialist.

  2. It should be possible to define a surjective morphism $\phi:\mathbb{A}^M\to G$ over $K$ (basically because a sphere is parametrizable). How small can we take $\deg \phi$ to be, in terms of $n$? Can we make sure it is substantially smaller than $\deg(G)$?

(Assume that the characteristic of $K$ is either $0$ or non-tiny, if it helps.)

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    $\begingroup$ What do you mean by $\deg(G)$? Asymptotics with respect to what? $\endgroup$
    – Sasha
    Commented Sep 12, 2021 at 19:11
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    $\begingroup$ The Cayley transform gives a nice parameterization. I would be surprised to find another of lower degree. $\endgroup$
    – Ben McKay
    Commented Sep 12, 2021 at 19:14
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    $\begingroup$ @Sasha Its degree as an affine variety. Asymptotics as $n\to \infty$. $\endgroup$
    – H A Helfgott
    Commented Sep 12, 2021 at 19:16
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    $\begingroup$ I would explicitly state Theorem 4.2 of that paper into the OP, since it converts question 1 into something like a dimer model counting question that should be attackable by existing methods in statistical mechanics without requiring any knowledge of algebraic geometry. (In particular, the answer is probably $\exp( (c+o(1)) n^2)$ for some constant $c$ that extremizes some entropy type functional.) $\endgroup$
    – Terry Tao
    Commented Sep 13, 2021 at 2:07
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    $\begingroup$ Question 2 cannot be answered as written, because there is no non-constant morphism from $\mathbb{A}^n$ to $SO(2) = \{ \left[ \begin{smallmatrix} x&y \\-y&x \end{smallmatrix} \right] : x^2+y^2=1 \}$. The issue is easiest to see if there is a square root of $-1$ in $K$. In this case, $(x+\sqrt{-1} y)(x-\sqrt{-1} y)=1$ on $SO(2)$, so $x+\sqrt{-1} y$ must pull back to a unit in the coordinate ring of $\mathbb{A}^n$, but there are no nontrivial units in $K[t_1, \ldots, t_n]$. $\endgroup$
    – David E Speyer
    Commented Sep 15, 2021 at 2:01

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