On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials

Sizhuo Yan; Jianting Yang

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arXiv · 2610.01804

On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials

Sizhuo Yan·Jianting Yang

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Abstract

Jurij Volčič conjectured that a noncommutative polynomial $g$ belongs to the two-sided ideal generated by $f_1,\ldots,f_l$ 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 $f_1,\ldots,f_l$. In this paper, we show this equivalence when the generators $f_1,\ldots,f_l$ are homogeneous. We then give an explicit counterexample in the non-homogeneous case.

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BibTeX RIS

Sizhuo Yan, Jianting Yang. 2026-10-01. On a Rank Nullstellensatz Conjecture for Noncommutative Polynomials. https://arxiv.org/abs/2610.01804

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