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

## TL;DR

This paper establishes optimal strong approximation results for quadratic forms over polynomial rings over finite fields, providing new bounds and methods that improve understanding of solutions to quadratic equations in this setting.

## Contribution

The paper introduces a stationary phase theorem over function fields and a notion of anisotropic cones, leading to optimal bounds for strong approximation on quadrics over $F_q[t]$ for $d extgreater=5$ variables.

## Key findings

- Proves strong approximation with optimal bounds for $d extgreater=5$ variables.
- Provides a new proof of bounded diameter for Ramanujan graphs without relying on the Ramanujan conjecture.
- Develops a stationary phase theorem and anisotropic cones in the function field context.

## Abstract

Suppose $q$ is a fixed odd prime power, $F(\vec{x})$ is a non-degenerate quadratic form over $\mathbb{F}_q[t]$ of discriminant $\Delta$ in $d\geq 5$ variables $\vec{x}$, and $f,g\in\mathbb{F}_q[t]$, $\boldsymbol{\lambda}\in\mathbb{F}_q[t]^d$. 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 $\vec{x}\equiv\boldsymbol{\lambda}\bmod g$. 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 $(2+\varepsilon)\log_{k-1}|G|+O_{\varepsilon}(1)$. In contrast to the $d=4$ case, our result is optimal for $d\geq 5$. 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.