Let \(X\) be a variety over a number field \(F\) such that the set \(X(F)\) of rational points is nonempty. Let \(S\) be a finite set of places in \(F\). The strong approximation for \(X\) off the set \(S\) means that the diagonal image of \(X(F)\) is dense in the space of \(S\)-adéles \(X({\mathbb A}_{F}^{S}).\) The \(S\)-adéles are the adéles with places in \(S\) omitted. If the property of strong approximation holds for \(X\) then it implies a local-global principle for the existence of integral points on models over the ring of \(S\)-integers of \(F.\) There is an obstruction called the Brauer-Manin obstruction to strong approximation off \(S. \)
In this paper, the authors investigate strong approximation for varieties \(X/F\) given by an equation \[ q(x_{1},\dots ,x_{n}) = p(t), \] where \(q\) is a quadratic form of rank \(n\) in \(n\geq 3\) variables and \(p(t)\) is a nonzero polynomial.