Math @ Duke

Publications [#320386] of Lillian B. Pierce
Papers Published
 Pierce, LB; Schindler, D; Wood, MM, Representations of integers by systems of three quadratic forms,
Proceedings of the London Mathematical Society, vol. 3 no. 113
(2016),
pp. 289344, London Mathematical Society [doi]
(last updated on 2018/07/20)
Abstract: 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; 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 $k \geq 10$ for "almost all" tuples, under appropriate nonsingularity assumptions on the forms $Q_1,Q_2,Q_3$. To accomplish this, we develop a threedimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.


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