Let \(\mathcal O=\mathbb F_q[t]\) be the polynomial ring over the finite field \(\mathbb F_q\) with \(q\) elements, where \(q\) is a fixed odd prime power, let \(F({\mathbf x})\) be a non-degenerate quadratic form of discriminant \(\Delta\) over \(\mathcal O\) in \(d\geq 4\) variables \(\mathbf{x}=(x_1,\ldots,x_d)\), and let \(f\in \mathcal O\). In this paper, the authors study the optimal strong approximation problem for the quadric \(X_f\) given by the equation \(F(\mathbf{x})=f\). That is, given \(g\in \mathcal O\) and \(\boldsymbol{\lambda} =(\lambda_1,\ldots,\lambda_d)\in \mathcal O^d\), they consider solutions \(\mathbf{x}=(x_1,\ldots,x_d)\in \mathcal O^d\) to the equation \(F(\mathbf{x})=f\) for which \(\mathbf{x}\equiv \boldsymbol{\lambda} \mbox{ mod }g\) (that is, \(x_i \equiv \lambda_i \mbox{ mod }g\) for all \(i=1,\ldots,d\)). Necessary local local conditions for such a solution are that \(X_f(\mathbb F_q((1/t)))\neq \emptyset\) and, for all irreducible \(\varpi \in \mathcal O\), \(F(\mathbf{x})=f\) has a solution \(\mathbf{x}_{\varpi}\in \mathcal O_{\varpi}^d\) such that \(\mathbf{x}_{\varpi} \equiv \boldsymbol{\lambda}\mbox{ mod }\varpi^{\mbox{ord}_{\varpi}(g)}\). The main strong approximation result obtained (Theorem 1.1) is as follows. Let \(\varepsilon >0\) be given, let \(f,g\) be polynomials in \(\mathcal O\) for which the irreducible divisors of \(\Delta\) appear with bounded multiplicity in \(fg\), and let \(\boldsymbol{ \lambda}\in \mathcal O^d\). Suppose that the necessary local conditions for a solution are satisfied. Then there is a constant \(C_{\varepsilon,F}\), independent of \(f\), \(g\) and \(\boldsymbol{\lambda}\), such that if \(d\geq 5\) and \(\mbox{deg }f \geq (4+\varepsilon)\mbox{deg }g+C_{\varepsilon,f}\), then there exists \(\mathbf{x}\in \mathcal O^d\) such that \(F(\mathbf{x})=f\) and \(\mathbf{x}\equiv \boldsymbol{\lambda}\mbox{ mod }g\). In the case \(d=4\), the same result holds with \((4+\varepsilon)\) replaced by \((6+\varepsilon)\).
The method of proof is based on a version of the circle method that was developed by D. R. Heath-Brown over the integers [J. Reine Angew. Math. 481, 149–206 (1996; Zbl 0857.11049)] and further developed in a paper of the first author [Duke Math. J. 168, 1887–1927 (2019; Zbl 1443.11030)] to prove an optimal strong approximation result for quadratic forms over the integers. In the present paper, the authors extend the circle method over function fields by proving a stationary phase theorem that allows them to bound certain oscillatory integrals that appear in the circle method. The result obtained is optimal for \(d\geq 5\).
The strong approximation result stated above is used to give a new proof, independent of the Ramanujan conjecture over function fields, that the diameter of a \(k\)-regular Morgenstern Ramanujan graph \(G\) is bounded above by \((2+\varepsilon)\log_{k-1}|G|+O_{\varepsilon}(1)\).