Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field
Abstract
We study the rational canonical form of sparse random matrices over a finite field. Suppose An∈Matn(𝔽p) has independent and αn-balanced entries. We prove that if nαn−log n→+∞, then, for every fixed collection of distinct monic irreducible polynomials over 𝔽p, the corresponding primary partitions of An converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform random matrix model. This strengthens the finite-field cokernel universality theorem of Lee (arXiv:2511.13070), which assumes αn=clog n/n for a fixed c>1, by resolving the additive endpoint case explicitly left open in his work. Moreover, our result extends universality from ordinary cokernels to the full rational canonical form. The condition is optimal for universality uniformly over fixed irreducible polynomials. Our proof uses the surjection moment method developed by Sawin and Wood (arXiv:2210.06279) over 𝔽p[t], together with a refinement of Lee's Fourier estimates that retains the full leading-order decay. We also construct degree-d critical sparse obstructions, suggesting a polynomial-dependent threshold of 1/d for irreducible polynomials of minimal degree d.
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BibTeX
@misc{shen2026sharp,
title = {{Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field}},
author = {Jiahe Shen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.01413},
eprint = {2609.01413},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01413},
}
RIS
TY - UNPB
TI - Sharp Threshold for Universality of Rational Canonical Forms over a Finite Field
AU - Jiahe Shen
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.01413
UR - https://arxiv.org/abs/2609.01413
AB - We study the rational canonical form of sparse random matrices over a finite field. Suppose A_n∈Mat_n(𝔽_p) has independent and α_n-balanced entries. We prove that if nα_n−log n→+∞, then, for every fixed collection of distinct monic irreducible polynomials over 𝔽_p, the corresponding primary partitions of A_n converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform random matrix model. This strengthens the finite-field cokernel universality theorem of Lee (arXiv:2511.13070), which assumes α_n=clog n/n for a fixed c>1, by resolving the additive endpoint case explicitly left open in his work. Moreover, our result extends universality from ordinary cokernels to the full rational canonical form. The condition is optimal for universality uniformly over fixed irreducible polynomials. Our proof uses the surjection moment method developed by Sawin and Wood (arXiv:2210.06279) over 𝔽_p[t], together with a refinement of Lee's Fourier estimates that retains the full leading-order decay. We also construct degree-d critical sparse obstructions, suggesting a polynomial-dependent threshold of 1/d for irreducible polynomials of minimal degree d.
ER -