Determinantal descent in dimension two: the pencil determinant is a multiple of $x_1x_2-\rho x_0^2$

Determinantal descent in dimension two: a singular pencil over the candidate's hull descends to the Diaz conic.

Let $K \subseteq \mathbb{C}$ be a subfield, $u \neq 0$ transcendental over $K$ with $\rho = u\bar u \in K$, and let $A, B, C$ be $2\times2$ matrices with entries in $K$. If

then there is $c \in K$ such that for all $x, y, z \in \mathbb{C}$

Why. Polarizing the $2\times2$ determinant writes $\det(xA+yB+zC)$ as a ternary quadratic form with six coefficients in $K$. Substituting $\bar u = \rho/u$ into the hypothesis and clearing $u^2$ turns it into the vanishing of a polynomial of degree four in $u$ with coefficients in $K$; transcendence of $u$ kills all five of its coefficients. What survives is $\det B = 0$, $\det C = 0$, the vanishing of the $xy$- and $xz$-coefficients, and $\det A + \rho\,\beta = 0$ where $\beta$ is the $yz$-coefficient. The displayed identity is then exactly what is left, with $c = \beta$.

Role. This is the $2\times2$ case of the manuscript's Determinantal descent of the Diaz conic, combined with the coefficient-matching half of Uniqueness of the $2\times2$ obstruction. It says that a singular matrix over the candidate's hull $W_u = \overline{\mathbb{Q}} \oplus \overline{\mathbb{Q}}u \oplus \overline{\mathbb{Q}}\bar u$ cannot be singular by accident: its determinant form, as a form in the three hull coordinates, is forced to be a scalar multiple of $\mathcal{Q}_\rho = x_1x_2 - \rho x_0^2$ — the conic on which the candidate lives. Two special cases are worth reading off: taking $x = z = 0$ gives $\det B = 0$, taking $x = y = 0$ gives $\det C = 0$; and $c = 0$ is exactly the degenerate branch in which the whole pencil is singular, which by the Dasgupta--Kakde alternative is the branch with a vanishing algebraic matrix coefficient.

Note that the statement is unconditional given the transcendence hypothesis — no unproved transcendence conjecture enters — because $\bar u = \rho/u$ makes the candidate's hull a rational function field in one transcendental.

Source: Carlo Perassi's private manuscript on Diaz's modulus conjecture, section The precise open boundary, theorem Determinantal descent of the Diaz conic (case $m = n = t = 2$) together with the coefficient comparison in the theorem Uniqueness of the $2\times2$ obstruction. No novelty is claimed.