The complete, two-orbit and cyclic shapes of a five-dimensional W ⊇ span(1, u, u⁻¹) in canonical form: D1, D2a or D2b

Let $u \in \mathbb{C}$ be transcendental, that is, $u \notin \overline{\mathbb{Q}}$ (the algebraic numbers); spans and linear independence are over $\overline{\mathbb{Q}}$. Let $W \subseteq \mathbb{C}$ be a $\overline{\mathbb{Q}}$-vector subspace of dimension $5$ with $1, u, u^{-1} \in W$, that is, $H_0 \subseteq W$ for $H_0 = \mathrm{span}(1, u, u^{-1})$. Let $P, Q \in \overline{\mathbb{Q}}[X]$ be coprime with $\max(\deg P, \deg Q) = 2$, write $P_c = P(u) - cQ(u)$, and suppose $W$ has one of the three shapes of DiazModulus.quadratic_complement_eigen_forms:

• (scalar) for some $c \in \overline{\mathbb{Q}}$, $Q(u)R(u)/(uP_c) \in W$ for every $R \in \overline{\mathbb{Q}}[X]$ of degree at most $2$;
• (two-orbit) for some $c_1 \neq c_2$ in $\overline{\mathbb{Q}}$ and $R_1, R_2 \in \overline{\mathbb{Q}}[X]$ of degree at most $2$, $W = \mathrm{span}(1, u, u^{-1}, Q(u)R_1(u)/(uP_{c_1}), Q(u)R_2(u)/(uP_{c_2}))$;
• (cyclic) for some $c \in \overline{\mathbb{Q}}$ and $R_0, R_1 \in \overline{\mathbb{Q}}[X]$ of degree at most $2$, $W = \mathrm{span}(1, u, u^{-1}, Q(u)R_0(u)/(uP_c), Q(u)(Q(u)R_0(u) + P_cR_1(u))/(uP_c^2))$.

Then one of the following holds:

• (D1) there are linearly independent $t_0, t_1, t_2 \in \mathbb{C}$ with $t_j \in W$ and $ut_j \in W$ for every $j$;
• (D2a, two-orbit) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), P'(u)S_2(u)/(uQ'(u)))$ for some $P', Q', S_1, S_2 \in \overline{\mathbb{Q}}[X]$ with $P', Q'$ coprime, $\max(\deg P', \deg Q') = 2$ and $\deg S_1, \deg S_2 \leq 2$;
• (D2b, cyclic) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), Q'(u)(Q'(u)S_1(u) + P'(u)S_2(u))/(uP'(u)^2))$ for some such $P', Q', S_1, S_2$.

Proof. By DiazModulus.coprime_pencil_mobius, $(P - cQ, Q)$ and, for $c_1 \neq c_2$, $(P - c_1Q, P - c_2Q)$ are again coprime of maximal degree $2$. Scalar: $t = (u^{-1}, 1, Q(u)/(uP_c))$ and $ut = (1, u, uQ(u)/(uP_c))$ lie in $W$ (take $R = 1$ and $R = X$). A relation $au^{-1} + b + eQ(u)/(uP_c) = 0$ times $uP_c$ gives $(a + bX)(P - cQ) + eQ = 0$ in $\overline{\mathbb{Q}}[X]$ ($u$ is transcendental), and coprimality with degree $2$ forces $a = b = e = 0$: (D1). Two-orbit: with $P' = P - c_1Q$, $Q' = P - c_2Q$, $d = c_1 - c_2$, $S_1 = R_1/d$, $S_2 = -R_2/d$, the canonical generators differ from the given ones by $S_1(u)/u, S_2(u)/u \in H_0$, so the spans agree: (D2a); the denominators are non-zero, the five generators of $W$ being independent. Cyclic: $P' = P - cQ$, $Q' = Q$, $S_1 = R_0$, $S_2 = R_1$ give (D2b) verbatim.

Novelty. Not found in the sources read; routine.