Counting solutions to the quadratic determinant equation
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916014095073280 |
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| author | Chapman, Jonathan Mudgal, Akshat |
| author_facet | Chapman, Jonathan Mudgal, Akshat |
| contents | Given $h, N \in \mathbb{N}$ satisfying $1 \leqslant h \leqslant N^2$, we prove an asymptotic formula for the number of solutions to the equation $x_1 x_2 - x_3 x_4 = h$ with $x_1, \ldots, x_4 \in [-N,N] \cap \mathbb{Z}$. We use a combination of combinatorial and analytic arguments in physical space along with bounds for Kloosterman sums. Our main result concerns the case when $h = N^2 + O(N)$, wherein we obtain square-root cancellation error terms by bypassing Kloosterman sum bounds and exploiting an additional symmetry available in this setting via Ramanujan sums. This confirms a speculation of Dhanda-Haynes-Prasala in a very general form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15434 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting solutions to the quadratic determinant equation Chapman, Jonathan Mudgal, Akshat Number Theory 11D45, 11D09, 11N37 Given $h, N \in \mathbb{N}$ satisfying $1 \leqslant h \leqslant N^2$, we prove an asymptotic formula for the number of solutions to the equation $x_1 x_2 - x_3 x_4 = h$ with $x_1, \ldots, x_4 \in [-N,N] \cap \mathbb{Z}$. We use a combination of combinatorial and analytic arguments in physical space along with bounds for Kloosterman sums. Our main result concerns the case when $h = N^2 + O(N)$, wherein we obtain square-root cancellation error terms by bypassing Kloosterman sum bounds and exploiting an additional symmetry available in this setting via Ramanujan sums. This confirms a speculation of Dhanda-Haynes-Prasala in a very general form. |
| title | Counting solutions to the quadratic determinant equation |
| topic | Number Theory 11D45, 11D09, 11N37 |
| url | https://arxiv.org/abs/2605.15434 |