Theorem E: the conjugation hull of a polynomial in u and ū carries a 2×3 configuration exactly when it is five-dimensional and in the polynomial, involution or real pencil class

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 $c_j \in \overline{\mathbb{Q}}$ for all $j \in \mathbb{Z}$, and put $z = \sum_{j=-k}^{k} c_jv^j$ and $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, z, \bar z)$. Then $W$ carries a $2 \times 3$ configuration (free $x_0, x_1$ and free $y_0, y_1, y_2$ over $\overline{\mathbb{Q}}$ with all six products $x_iy_j$ in the space) if and only if $\dim_{\overline{\mathbb{Q}}} W = 5$ and $W$ is in one of the following 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 $\deg G \geq 2$.

Since $u = sv$ and $\bar u = s/v$, the numbers $z$ are exactly the polynomials in $u$ and $\bar u$ with algebraic coefficients, and $W$ is the conjugation hull of $z$. The three classes are pairwise disjoint (DiazModulus.laurent_conj_hull_configuration_classes_disjoint), and each forces $\dim W = 5$ (DiazModulus.laurent_conj_hull_class_finrank_eq_five).

Proof. As $s$ is real, $v = u/s$ has $v\bar v = 1$ and $v \notin \overline{\mathbb{Q}}$, and $\mathrm{span}(1, u, \bar u, A, B) = \mathrm{span}(1, v, \bar v, A, B)$ for all $A, B$; this moves every node between $u$ and $v$. Only if: a configuration forces $\dim W > 4$ by DiazModulus.laurent_conj_hull_finrank_le_four_no_two_by_three_configuration, and $\dim W \leq 5$ by counting generators; if $W$ contains some $v^2P(v)$ with $P \neq 0$, DiazModulus.laurent_conj_hull_with_polynomial_part_configuration_class gives $\alpha v^2 + \beta v^3 \in W$, which is (P) after rescaling by $s^2$ and $s^3$, and otherwise DiazModulus.laurent_conj_hull_without_polynomial_part_configuration_classes gives (Q) or (R). If: in class (P), $W$ is conjugation-stable, so it contains the hull of $P(u) = \alpha u^2 + \beta u^3$, which carries a configuration by Theorem C (DiazModulus.polynomial_hull_two_by_three_configuration_iff, degree $2$ or $3$). Classes (Q) and (R) carry the configurations of DiazModulus.laurent_involution_space_carries_two_by_three_configuration and DiazModulus.laurent_real_pencil_normal_form_carries_two_by_three_configuration.

Novelty. Not found in the sources read, and not routine. The "only if" half, the classification, was not found in the sources read; its method is partly that of Bachoc, Serra and Zémor (2017, §7, and Lemma 4). The "if" half consists of printed configurations or substitutions in them: class (P) is Diaz (2007, Th. 7(1)), already in Diaz (2004, Th. 2); class (Q) is Diaz (2004, Th. 2) at $x = (1, (u + \beta\bar u)/s)$; class (R) consists of the real pencil spaces of DiazModulus.conj_stable_real_pencil_space_carries_two_by_three_configuration.