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On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials
Abstract
Jurij Volčič conjectured that a noncommutative polynomial g belongs to the two-sided ideal generated by f1,…,fl if and only if, for matrices of every size, the rank of the evaluation of g is bounded by a constant times the maximum rank of the evaluations of f1,…,fl. In this paper, we show this equivalence when the generators f1,…,fl are homogeneous. We then give an explicit counterexample in the non-homogeneous case.
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BibTeX
@misc{yan2026rank,
title = {{On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials}},
author = {Sizhuo Yan and Jianting Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01804},
eprint = {2610.01804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01804},
}
RIS
TY - UNPB
TI - On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials
AU - Sizhuo Yan
AU - Jianting Yang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01804
UR - https://arxiv.org/abs/2610.01804
AB - Jurij Volčič conjectured that a noncommutative polynomial g belongs to the two-sided ideal generated by f_1,…,f_l if and only if, for matrices of every size, the rank of the evaluation of g is bounded by a constant times the maximum rank of the evaluations of f_1,…,f_l. In this paper, we show this equivalence when the generators f_1,…,f_l are homogeneous. We then give an explicit counterexample in the non-homogeneous case.
ER -