Singular Cholesky Fibers over Finite Fields
Abstract
For a finite field Fq, consider the triangular Cholesky map Γn,q(U)=UTU from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over F2 we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying UTU=0. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every F2e, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to O(n2) integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.
Related
- Maximal singular curves over finite fields from elliptic and hyperelliptic curves
- Finite Directional Coefficient Modules over Euler Fibers of A-Hypergeometric Systems
- Rational Points on Danielewski Surfaces over Finite fields and Polygonal Numbers
- Post-critically finite polynomials over cyclotomic closures of number fields
- Lie Algebras of Locally Finite Derivations over Fields of Positive Characteristic
BibTeX
@misc{wu2026singular,
title = {{Singular Cholesky Fibers over Finite Fields}},
author = {Hongfeng Wu and Kai Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00533},
eprint = {2609.00533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00533},
}
RIS
TY - UNPB
TI - Singular Cholesky Fibers over Finite Fields
AU - Hongfeng Wu
AU - Kai Zhou
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00533
UR - https://arxiv.org/abs/2609.00533
AB - For a finite field F_q, consider the triangular Cholesky map Γ_n,q(U)=U^TU from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over F_2 we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying U^TU=0. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every F_2^e, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to O(n^2) integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.
ER -