PreprintCC BY 4.0arXiv
Counting Representations of Quadratic Forms
Abstract
We answer a question of C. S. Herz about the number of integral k× m matrices T such that TtT≤ R for a fixed positive definite m× m matrix R. We give an asymptotic formula for the count. This can be seen as a generalization of the Gauss circle problem, as counting representation of quadratic forms by the identity Ik, and as counting integral points bounded by the Stiefel manifold TtT=R. The main tool is a bound for Bessel functions of matrix argument that was proved over Jordan algebras.
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BibTeX
@misc{felber2026counting,
title = {{Counting Representations of Quadratic Forms}},
author = {Gilles Felber},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07133},
eprint = {2609.07133},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07133},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Counting Representations of Quadratic Forms
AU - Gilles Felber
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07133
UR - https://arxiv.org/abs/2609.07133
AB - We answer a question of C. S. Herz about the number of integral k× m matrices T such that T^tT≤ R for a fixed positive definite m× m matrix R. We give an asymptotic formula for the count. This can be seen as a generalization of the Gauss circle problem, as counting representation of quadratic forms by the identity I_k, and as counting integral points bounded by the Stiefel manifold T^tT=R. The main tool is a bound for Bessel functions of matrix argument that was proved over Jordan algebras.
ER -