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 forms Q(1) ,..., Q(R) in 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 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.
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