Systems of rational quadratic forms (Q1881530)

Let \(Q_1,\dots, Q_r\in\mathbb{Q}[X_1,\dots, X_s]\) be quadratic forms. It was shown by \textit{W. M. Schmidt} [Theorie des nombres, Semin. Delange-Pisot-Poitou, Paris 1980--81, Prog. Math. 22, 281--307 (1982; Zbl 0492.10017)] that the number of simultaneous integral solutions of size at most \(p\) is \(cP^{s-2r}+ o(P^{s-2r})\) providing that every form in the rational pencil has rank exceeding \(2r^2+ 3r\). The proof used the circle method, and \(c\) is the usual product of local densities. However further conditions are needed to ensure that \(c> 0\). Schmidt showed that for this it is sufficient that there is a non-singular real point, and that every form in the complex pencil has rank exceeding \(4r^2+ 4r\). He also showed that this last condition holds providing that every form in the rational pencil has rank exceeding \(4r^3+ 4r^2\). The present paper weakens these last two conditions, showing that \(c> 0\) if there is a non-singular real zero, and if either every form in the complex pencil has rank at least \(2r^2+ 3r\), or every form in the rational pencil has rank at least \(4r[(r^2+ 1)/2]\). Naturally the proof depends on proving the existence of non-singular \(p\)-adic zeros, and here it is shown that it suffices for every form in the \(p\)-adic pencil to have rank exceeding \(4[(r^2 +1)/2]\).