## 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 $(n_1,\ldots,n_R)$ 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 three-dimensional analogue of Kloosterman's circle method, in particular capitalizing on geometric properties of appropriate systems of three quadratic forms.

### Duke Scholars

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## Related Subject Headings

- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics

### Citation

*Proceedings of the London Mathematical Society*,

*3*(113), 289–344.

*Proceedings of the London Mathematical Society*3, no. 113 (2016): 289–344.

*Proceedings of the London Mathematical Society*, vol. 3, no. 113, London Mathematical Society, 2016, pp. 289–344.

## Published In

## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Publisher

## Related Subject Headings

- 4904 Pure mathematics
- 0104 Statistics
- 0101 Pure Mathematics