Arithmetic Conditions for the Invertibility of Circulant Matrices over Finite Fields
Abstract
This paper investigates arithmetic conditions, over finite fields, for the invertibility of circulant matrices whose first row is (a0,…,an−1), in terms of linear constraints on the entries a0,…,an−1. The coefficients of such linear constraints are determined by field traces arising from Frobenius orbits of roots of unity, and under additional assumptions they are given by Ramanujan sums reduced modulo the characteristic of the given field. These results provide finite-field analogues of the corresponding criteria over ℚ, where the relevant Ramanujan-sum linear constraints can be formulated without such additional assumptions, while also showing some differences caused by reduction modulo the characteristic. The constraints can often determine invertibility using aggregate information about the entries of a row, rather than the particular pattern of the entries. We also discuss the polynomial gcd criterion, which gives necessary and sufficient conditions for invertibility and yields further consequences for special families of circulant matrices.
Related
- Invertibility of structured perturbations of singular matrices over unital rings
- Rank and Invertibility of Dense Signed Random Regular Matrices
- On properties of the conformal circulant matrices
- Freeness for the G-circulant Decomposition of the Partial Transpose of Random Matrices
- High-Girth QLDPC Codes from Circulant and Affine Permutation Matrices
BibTeX
@misc{choi2026arithmetic,
title = {{Arithmetic Conditions for the Invertibility of Circulant Matrices over Finite Fields}},
author = {Jeong-Ok Choi and Youngmi Hur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30830},
eprint = {2610.30830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30830},
}
RIS
TY - UNPB
TI - Arithmetic Conditions for the Invertibility of Circulant Matrices over Finite Fields
AU - Jeong-Ok Choi
AU - Youngmi Hur
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30830
UR - https://arxiv.org/abs/2610.30830
AB - This paper investigates arithmetic conditions, over finite fields, for the invertibility of circulant matrices whose first row is (a_0,…,a_n−1), in terms of linear constraints on the entries a_0,…,a_n−1. The coefficients of such linear constraints are determined by field traces arising from Frobenius orbits of roots of unity, and under additional assumptions they are given by Ramanujan sums reduced modulo the characteristic of the given field. These results provide finite-field analogues of the corresponding criteria over ℚ, where the relevant Ramanujan-sum linear constraints can be formulated without such additional assumptions, while also showing some differences caused by reduction modulo the characteristic. The constraints can often determine invertibility using aggregate information about the entries of a row, rather than the particular pattern of the entries. We also discuss the polynomial gcd criterion, which gives necessary and sufficient conditions for invertibility and yields further consequences for special families of circulant matrices.
ER -