Optimal strong approximation for quadrics over ${\mathbb{F}}_q[t]$

Suppose $q$ is a fixed odd prime power, $F(\vec{x})$ is a non-degenerate quadratic form over ${\mathbb{F}}_q[t]$ of discriminant $\mathrm{\Delta }$ in $d\ge 5$ variables $\vec{x}$, and $f,g\in {\mathbb{F}}_q[t]$, $\mathit{\lambda }\in {\mathbb{F}}_q[t{]}^d$. We show that whenever $$\text{deg}$ f\geq (4+\varepsilon)$\text{deg}$ g+O_{\varepsilon,F}(1)$,$\gcd(\Delta^{\infty},fg)=O(1)$, and the necessary local conditions are satisfied, we have a solution $\vec{x}\in {\mathbb{F}}_q[t{]}^d$ to $F(\vec{x})=f$ such that $\vec{x}\equiv \mathit{\lambda }g$. For $d=4$, we show that the same conclusion holds if we instead have $$\text{deg}$ f\geq (6+\varepsilon)$\text{deg}$ g+O_{\varepsilon,F}(1)$. This gives us a new proof (independent of the Ramanujan conjecture over function fields proved by Drinfeld) that the diameter of any $k$-regular Morgenstern Ramanujan graphs $G$ is at most $(2+\epsilon ){\log }_{k-1}\mathrm{\mid }G\mathrm{\mid }+O_{\epsilon }(1)$. In contrast to the $d=4$ case, our result is optimal for $d\ge 5$. Our main new contributions are a stationary phase theorem over function fields for bounding oscillatory integrals, and a notion of anisotropic cones to circumvent isotropic phenomena in the function field setting.