Solving quadratic forms in restricted variables with the circle method

Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give conditions on both $\mathcal A$ and $f$ such that we can use the circle method to count such solutions of bounded height.

Publication Details

Journal
ArXiv.org
Published
2025-05-23
Primary Topic
Autobiographical and Biographical Writing
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article
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article

Solving quadratic forms in restricted variables with the circle method

ArXiv.org
Autobiographical and Biographical Writing
article

Solving quadratic forms in restricted variables with the circle method

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Abstract

Let $f(\mathbf x)$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal A$ we study the number of weighted solutions to $f(\mathbf x) = t$. In particular, we give conditions on both $\mathcal A$ and $f$ such that we can use the circle method to count such solutions of bounded height.

ArXiv.org
Reduced inequalities
Openalex Percentile: Top 45%
Autobiographical and Biographical Writing
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Solving quadratic forms in restricted variables with the circle method · ArXiv.org (2025) | TGRS Research Map | TGRS