Effective Hasse principle for the intersection of two quadrics

Abstract

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.