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  • 1
    Publication Date: 2016-09-03
    Description: It is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers $(n_1,\ldots ,n_R)$ by a system of quadratic forms $Q_1,\ldots , Q_R$ 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 $k \geq 10$ 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.
    Print ISSN: 0024-6115
    Electronic ISSN: 1460-244X
    Topics: Mathematics
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