The following theorem by Legendre is well-known: The integral ternary quadratic form $f(x,y,z) = ax^2 + by^2 +cz^2 \in {\bf Z}[x]$ represents $0$ non-trivially if and only if
- $a$, $b$, and $c$ do not have the same sign.
- $-bc$ is a quadratic residue $\rm mod|a|$, $-ca$ is a quadratic residue $\rm mod|b|$, and $-ab$ is a quadratic residue $\rm mod|c|$.
My question: Is an analogous theorem true if we replace $\bf Z$ by polynomial rings? Like ${\bf Z}[t]$ (so that $a(t)$, $b(t)$, $c(t)$ are polynomials with integral coefficients as are $x(t)$, $y(t)$, and $z(t)$) or by ${\bf F}_q[t]$ where ${\bf F}_q$ is the finite field of cardinality $q$? Does anyone know of a reference where this theorem has been proved? Thanks!