A conjugation hull in the polynomial, involution or real pencil class is five-dimensional

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ and $s \in \overline{\mathbb{Q}}$ with $s^2 = u\bar u$ (so $s = \pm|u|$ is real), and put $v = u/s$, so that $\bar v = 1/v$. Let $k \geq 0$, let $c_j \in \mathbb{C}$ for all $j \in \mathbb{Z}$ (they need not be algebraic), and put $z = \sum_{j=-k}^{k} c_jv^j$ and $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, z, \bar z)$. If $W$ is in one of the classes

• (P) $\alpha u^2 + \beta u^3 \in W$ for some $\alpha, \beta \in \overline{\mathbb{Q}}$, not both zero;
• (Q) $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, \bar v^2 + ev^2, \beta\bar v^3 + \zeta v^2 + ev^3)$ for some $e, \beta, \zeta \in \overline{\mathbb{Q}}$ with $e\bar e = 1$, $\beta\bar\beta = 1$ and $\zeta = -\beta e^2\bar\zeta$;
• (R) $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, g, \bar g)$ with $g = G(v) + \kappa\,\overline{vG(v)}$, for some $\kappa \in \overline{\mathbb{Q}}$ with $\kappa\bar\kappa = 1$ and some $G \in \overline{\mathbb{Q}}[X]$ with $G(0) = 0$ and $\deg G \geq 2$,

then $\dim_{\overline{\mathbb{Q}}} W = 5$.

Proof. Work in $v = u/s$, which is transcendental with $\bar v = 1/v$ since $s$ is real. In class (P), with $m = \alpha u^2 + \beta u^3 \in W$, also $\bar m \in W$, as the hull is conjugation-stable; $1, u, \bar u, m, \bar m$ are free, because a relation cleared by $v^3$ is a polynomial of degree $6$ in $v$ whose seven coefficients vanish, and $W$ has five generators. In class (Q) the normal-form generators are free in the same way, using $\beta \neq 0$ from $\beta\bar\beta = 1$. In class (R), with $n = \deg G \geq 2$ and $H = \sum\bar G_iX^{n-i}$, so that $H(v) = v^n\overline{G(v)}$ and $H(0) = \bar G_n \neq 0$, a relation cleared by $v^{n+1}$ and evaluated at $0$ loses its $g$ coordinate, its conjugate loses the $\bar g$ one, and $1, u, \bar u$ are free.

Novelty. Not found in the sources read; routine.