For u transcendental over a subfield K of ℂ with uū ∈ K, every p×q configuration over K in K + Ku + Kū has p + q ≤ 4

A $p \times q$ configuration over a field $K$ in a $K$-subspace $V \subseteq \mathbb{C}$ is a pair $x_1, \dots, x_p$ and $y_1, \dots, y_q$ of complex numbers, each family linearly independent over $K$, with every product $x_iy_j$ in $V$; it is the input of Roy's strong six exponentials theorem when $p = 2$, $q = 3$, $K = \overline{\mathbb{Q}}$ and $V = \widetilde{\mathcal{L}}$.

Let $K$ be a subfield of $\mathbb{C}$ and $u \in \mathbb{C}$ transcendental over $K$ with $\rho = u\bar u \in K$, and put $H_0 = K + Ku + K\bar u$ (note $\bar u = \rho/u$). Every $p \times q$ configuration over $K$ in $H_0$ (with $p, q \ge 1$) has $p + q \le 4$. In particular $H_0$ carries no $2 \times 3$ configuration, over any base field.

Proof. Since $\bar u = \rho/u$, $H_0$ is a subspace of $K(u)$ of dimension at most $3$, and DiazModulus.rank_one_config_card_le_finrank_add_one gives $p + q \le 4$.

Novelty. Tier 2 (short). Over $\overline{\mathbb{Q}}$ the absence of $2 \times 3$ configurations is a case of Roy's lemma on spaces of dimension three (Waldschmidt, DALAG, Lemma 12.16).