10

(Working in ZFC.)
No (re $\mathcal{L}_{\omega_1,\omega}$). Suppose it is. Consider the signature $\Sigma$ with just one binary relation symbol $<$. Let $\Sigma',\eta$ witness SED-ness for $\Sigma$.
Let $\pi:M\to V_\eta$ be elementary,with $\eta$ some sufficiently large limit ordinal, $M$ transitive, $M$ of cardinality $\kappa=2^{\aleph_0}$, with $\mathbb{...

8

(Working in ZFC.)
$\omega_2$ is not Fraissean. In fact, it is not Fraissean with respect to $\Sigma$, where $\Sigma$ is the signature with a single binary relation $<$.
To see this we use a variant of the argument you linked in the question. Suppose otherwise, and let $\Sigma'$ and $\eta$ witness this. Let $\gamma$ be a large enough ordinal and let $\pi:M\...

6

It depends on what diamond sequence you have.
$V=L$ proves there is a $\diamondsuit$-sequence of $L$-rank $\omega_1+1$: recall the famous way of proving the existence of $\diamondsuit$-sequence. The definition goes as follows (for example, Theorem 13.21 of Jech 3rd):
$(S_\alpha,C_\alpha)$ is the $<_L$-least pair $S_\alpha\subseteq\alpha$, $C_\alpha\...

6

Um, for each variable $z_k = x_k + iy_k$ we throw in the equation $x_k^2 + y_k^2 = 1$ and rewrite everything in terms of $x_k$ and $y_k$. I am missing something, I think.

5

If your main complexity objective is to keep the number of variables down, you can use a rational parametrization of the circle: for each variable $z_k\in S^1$, introduce a real variable $t_k$ and rewrite everything using the substitution
$$z_k=\frac{i-t_k}{i+t_k}=\frac{1-t_k^2}{1+t_k^2}+i\frac{2t_k}{1+t_k^2}.$$
(You will have to deal with $z_k=-1$ somehow.)

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