Low-rank matrices with three prescribed off-diagonal values over 𝔽_p

Abstract

Let p be a prime such that $2$ is a primitive root modulo p. We prove that, for every three-element set L⊆𝔽_p and every integer n≥1, there exists a symmetric n× n matrix over 𝔽_p with zero diagonal, off-diagonal entries in L, and rank O_p,L(n^1/3). This answers a question of Bukh for all such primes and, in particular, settles the case L={1,4,6}⊆𝔽₁₉ singled out on his problem list. When 0∉ L, a standard polynomial argument shows that the exponent 1/3 is optimal. The proof combines a geometric construction in characteristic two with an incidence matrix factorization over 𝔽_p. The same construction yields integer lifts L' of the prescribed sets L and symmetric L'-matrices of rational rank O_p,L(n^1/3). When 0∉ L', this improves Bukh's general quadratic lower bound, giving matrices of order Ω_p,L(r³) and rational rank at most r.

Projections in reduced group C*-algebras with prescribed trace values in (0,1)
Guoliang Yu
arXiv26 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theory Analysis & PDEsarXiv
The Karcher Mean on 2× 2 Triangular Matrices with Positive Diagonal Entries
Huajun Huang
arXiv29 Sep 2026PreprintCC BY-NC-SA 4.0Full textAlgebra & number theory Analysis & PDEsarXiv
Finite-Rank Lie Algebroids for Singular Foliations of Prescribed Vanishing Order
Austin Davis, Cheng-Yi Hung, Sangjun Ko +1
arXiv20 Oct 2026PreprintCC BY 4.0Full textGeometry & topologyarXiv
A note on near alternating sign matrices with prescribed row and column weights
Tommaso Traetta
arXiv7 Sep 2026PreprintCC BY-NC-ND 4.0Full textCombinatorics & discrete mathematicsarXiv
The Maximum Diagonal Product of a Stochastic Matrix of Bounded Rank
Yair Lavi
arXiv1 Nov 2026PreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{rakhmonov2026lowrank,
  title = {{Low-rank matrices with three prescribed off-diagonal values over $\mathbbF_p$}},
  author = {Firdavs Rakhmonov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39908},
  eprint = {2610.39908},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39908},
}

RIS

TY  - UNPB
TI  - Low-rank matrices with three prescribed off-diagonal values over 𝔽_p
AU  - Firdavs Rakhmonov
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39908
UR  - https://arxiv.org/abs/2610.39908
AB  - Let p be a prime such that $2$ is a primitive root modulo p. We prove that, for every three-element set L⊆𝔽_p and every integer n≥1, there exists a symmetric n× n matrix over 𝔽_p with zero diagonal, off-diagonal entries in L, and rank O_p,L(n^1/3). This answers a question of Bukh for all such primes and, in particular, settles the case L={1,4,6}⊆𝔽_19 singled out on his problem list. When 0∉ L, a standard polynomial argument shows that the exponent 1/3 is optimal. The proof combines a geometric construction in characteristic two with an incidence matrix factorization over 𝔽_p. The same construction yields integer lifts L' of the prescribed sets L and symmetric L'-matrices of rational rank O_p,L(n^1/3). When 0∉ L', this improves Bukh's general quadratic lower bound, giving matrices of order Ω_p,L(r^3) and rational rank at most r.
ER  -

Abstract

Related

Cite