If a space of dimension ≤ 5 containing 1, u, 1/u carries a 2×3 configuration x ⊗ y with x₁/x₀ = p(u)/q(u) in lowest terms, then deg p, deg q ≤ 2

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ and let $W \subseteq \mathbb{C}$ be a finite-dimensional $\overline{\mathbb{Q}}$-subspace with $\dim W \leq 5$ and $1, u, u^{-1} \in W$. Let $x_0, x_1$ be linearly independent over $\overline{\mathbb{Q}}$, and $y_0, y_1, y_2$ too, with $x_iy_j \in W$ for all $i, j$, and let $p, q \in \overline{\mathbb{Q}}[X]$ be coprime with $\max(\deg p, \deg q) \geq 1$ and $x_1q(u) = x_0p(u)$. Then $\max(\deg p, \deg q) \leq 2$.

Proof. Coprimality and $\max(\deg p, \deg q) \geq 1$ give $q(u) \neq 0$, so $x_1 = x_0h$ with $h = p(u)/q(u)$; as $x$ is free, $h \notin \overline{\mathbb{Q}}$, so $h$ is transcendental over $\overline{\mathbb{Q}}$. Put $Y' = \mathrm{span}(x_0y_0, x_0y_1, x_0y_2)$ and $XY = Y' + hY' \subseteq W$, the span of the six products, so that $\dim(Y' \cap hY') \geq 1$. If $H = \mathrm{span}(1, u, u^{-1}) \subseteq XY$, a non-zero $ht \in Y' \cap hY'$ with $t \in Y'$ and some $c' \in Y'$ outside $\mathrm{span}(t, ht)$ give $XY \subseteq \mathrm{span}(t, th, th^2, tk, thk)$ with $k = c'/t$, and DiazModulus.progression_pair_span_ratio_degree_le_two applies. Otherwise $\dim XY \leq 4$, so two planes in $Y'$ meet and give $t, th, th^2 \in Y'$; then $XY = \mathrm{span}(t, th, th^2, th^3)$, the span of $H$ and $XY$ has dimension $5$, and DiazModulus.progression_space_ratio_degree_le_two applies.

Novelty. Not found in the sources read; routine. When the six products span four dimensions, the step to a progression $t, th, th^2, th^3$ is that of Bachoc, Serra and Zémor (2017, Lemma 4).