Published online by Cambridge University Press: 26 August 2016
We consider a smooth system of two homogeneous quadratic equations over $\mathbb{Q}$ in
$n\geqslant 13$ variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over
$\mathbb{R}$. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.