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It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers    (   n  1   ,  …  ,   n  R   )    by a system of quadratic formsQ  1   ,  …  ,   Q  Rin   k   variables, as long as   k   is sufficiently large with respect to   R  ; reducing the required number of variables remains a significant open problem. In this work, we consider the case of three forms and improve on the classical result by reducing the number of required variables to    k  ⩾  10    for ‘almost all’ tuples, under a non‐singularity assumption on the formsQ  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.

Representations of integers by systems of three quadratic forms