A nowhere-zero point for several linear mappings simultaneously
Abstract
Let q=pk, and let A1,…,Ar−1 be invertible n× n matrices over 𝔽q. We prove that, if k≥ r, there is a vector x for which x,A1x,…,Ar−1x are all nowhere zero. For r=2 this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomial in the product of the coordinate forms. Frobenius powers keep every exponent below pr, and finite-field polynomial nonvanishing supplies the required vector. The same method treats rectangular matrices with independent rows and prescribed forbidden values. The method also gives an extension-degree criterion for representable matroids and clarifies an unpublished prime-field conjecture of M. J. Moghaddamzadeh. Projective-geometric examples explain why the analogous field-size statement fails over proper extensions and, translated back to matrices, give lower bounds for the large-field problem.
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BibTeX
@misc{jafari2026nowherezero,
title = {{A nowhere-zero point for several linear mappings simultaneously}},
author = {Amir Jafari},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.23647},
eprint = {2608.23647},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.23647},
}
RIS
TY - UNPB
TI - A nowhere-zero point for several linear mappings simultaneously
AU - Amir Jafari
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2608.23647
UR - https://arxiv.org/abs/2608.23647
AB - Let q=p^k, and let A_1,…,A_r−1 be invertible n× n matrices over 𝔽_q. We prove that, if k≥ r, there is a vector x for which x,A_1x,…,A_r−1x are all nowhere zero. For r=2 this recovers the theorem of Alon and Tarsi over nonprime finite fields. The proof tracks one monomial in the product of the coordinate forms. Frobenius powers keep every exponent below p^r, and finite-field polynomial nonvanishing supplies the required vector. The same method treats rectangular matrices with independent rows and prescribed forbidden values. The method also gives an extension-degree criterion for representable matroids and clarifies an unpublished prime-field conjecture of M. J. Moghaddamzadeh. Projective-geometric examples explain why the analogous field-size statement fails over proper extensions and, translated back to matrices, give lower bounds for the large-field problem.
ER -