Representations of integers by systems of three quadratic forms

It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\dots ,n_R)$ by a system of quadratic forms $Q_1,\dots ,Q_R$ in $k$ variables, as long as $k$ is sufficiently large; reducing the required number of variables remains a significant open problem. In this work, we consider the case of 3 forms and improve on the classical result by reducing the number of required variables to $k\ge 10$ for “almost all” tuples, under appropriate nonsingularity assumptions on the forms $Q_1,Q_2,Q_3$. To accomplish this, we develop a three-dimensional analogue of Kloosterman’s circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.