For a ratio h of degree 2, pole fractions q(αq + βJ)/(u(p − λq)) at three distinct λ are linearly independent modulo span(1, u, u⁻¹)

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 $p, q \in \overline{\mathbb{Q}}[X]$ be coprime with $\max(\deg p, \deg q) = 2$, and $J = p'q - pq'$. Let $\lambda_0, \lambda_1, \lambda_2 \in \overline{\mathbb{Q}}$ be distinct, let $\alpha_i, \beta_i \in \overline{\mathbb{Q}}$ with $(\alpha_i, \beta_i) \neq (0, 0)$ for each $i$, and put $g_i = q(u)(\alpha_iq(u) + \beta_iJ(u))/(u(p(u) - \lambda_iq(u)))$. If $a, b, c, d_0, d_1, d_2 \in \overline{\mathbb{Q}}$ satisfy $a + bu + cu^{-1} + d_0g_0 + d_1g_1 + d_2g_2 = 0$, then $a = b = c = 0$ and $d_0 = d_1 = d_2 = 0$.

So $1, u, u^{-1}, g_0, g_1, g_2$ are linearly independent: pole fractions at three distinct values of $\lambda$ are independent modulo $H_0 = \mathrm{span}(1, u, u^{-1})$.

Proof. The denominators are non-zero: $u \neq 0$, and $p - \lambda q \neq 0$ by coprimality and degree, so its value at the transcendental $u$ is not $0$. Write $h = p(u)/q(u)$ and $w = J(u)/q(u)$, so that $g_i = (q(u)/u)(\alpha_i + \beta_iw)/(h - \lambda_i)$. The $H_0$-part $a + bu + cu^{-1}$ is $R(u)/u$ with $R = c + aX + bX^2$, and DiazModulus.quadratic_pencil_wronskian_span writes $R = a'q + b'p + c'J$. Dividing the relation by $q(u)/u$ and multiplying by $\prod_j(h - \lambda_j)$ gives $A(h) + B(h)w = 0$ with $A = (a' + b'X)\Pi + \sum_id_i\alpha_i\Pi_i$ and $B = c'\Pi + \sum_id_i\beta_i\Pi_i$, where $\Pi = \prod_j(X - \lambda_j)$ and $\Pi_i = \prod_{j \neq i}(X - \lambda_j)$. Then DiazModulus.quadratic_ratio_wronskian_free gives $A = B = 0$. Evaluating at $X = \lambda_i$, where $\Pi_i(\lambda_i) \neq 0$ as the $\lambda_j$ are distinct and every other term vanishes, leaves $d_i\alpha_i = d_i\beta_i = 0$, so $d_i = 0$. Then $a + bu + cu^{-1} = 0$, and $a = b = c = 0$ because $c + aX + bX^2$ vanishes at $u$.

Novelty. Not found in the sources read; routine.