Effective Hasse principle for the intersection of two quadrics

TL;DR

This paper provides an explicit algorithm to determine the existence of rational solutions for a smooth intersection of two quadrics in at least 13 variables, building on the known validity of the Hasse principle in this setting.

Contribution

It introduces a practical algorithm to decide and compute rational solutions for the intersection of two quadrics, extending theoretical results into computational methods.

Findings

Algorithm successfully determines the existence of rational solutions.

Computes explicit solutions when they exist.

Builds on Mordell's classical results from 1959.

Abstract

We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 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 the reals. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists, and if so, to compute one.