Corollary E(b): for z ≡ Aū³ + Bū² + Cu² + Du³ with a five-dimensional hull, a 2×3 configuration exists iff NΔ(|Δ|² − |u|²N²) = 0, N = |A|² − |D|², Δ = AC̄ − BD̄

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 $A, B, C, D \in \overline{\mathbb{Q}}$, let $z \in \mathbb{C}$ with $z - (A\bar u^3 + B\bar u^2 + Cu^2 + Du^3) \in \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u)$, and let $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, z, \bar z)$ with $\dim_{\overline{\mathbb{Q}}} W = 5$. Put $\Delta = A\bar C - B\bar D$ and $N = A\bar A - D\bar D$, and consider 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:

• $W$ is in class (P) if and only if $\Delta = 0$;
• $W$ is in class (Q) if and only if $N = 0$ and $(A, D) \neq (0, 0)$;
• $W$ is in class (R) if and only if $|\Delta|^2 = s^2N^2$ and $N \neq 0$;
• $W$ 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 $N\Delta\,(|\Delta|^2 - s^2N^2) = 0$.

Here $s^2 = |u|^2$.

Proof. $s$ is real since $s^2 = |u|^2 > 0$, so $v = u/s$ has $v\bar v = 1$ and $v \notin \overline{\mathbb{Q}}$, $\mathrm{span}(1, u, \bar u, \cdot, \cdot) = \mathrm{span}(1, v, \bar v, \cdot, \cdot)$, and $z \equiv a\bar v^3 + b\bar v^2 + cv^2 + dv^3$ with $a = As^3$, $b = Bs^2$, $c = Cs^2$, $d = Ds^3$, so that $a\bar c - b\bar d = s^5\Delta$ and $a\bar a - d\bar d = s^6N$. The three classes are then read off DiazModulus.cubic_window_hull_polynomial_class_iff, DiazModulus.cubic_window_hull_involution_class_iff and DiazModulus.cubic_window_hull_real_pencil_class_iff (for (P), $\alpha u^2 + \beta u^3 = (\alpha s^2)v^2 + (\beta s^3)v^3$). Writing $z = \sum_{|j| \leq 3} c_jv^j$, Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff, $k = 3$) says that $W$ carries a configuration exactly when it is in one of the classes, that is $\Delta = 0$, or $N = 0$ with $(A, D) \neq (0, 0)$, or $|\Delta|^2 = s^2N^2$ with $N \neq 0$; since $A = D = 0$ forces $\Delta = 0$, this is $N\Delta(|\Delta|^2 - s^2N^2) = 0$.

Novelty. Not found in the sources read; routine, given Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff).