A Homogeneous Nullstellensatz for Joint Invariant Subspaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917294806925312 |
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| author | Yan, Sizhuo Yang, Jianting Zhi, Lihong |
| author_facet | Yan, Sizhuo Yang, Jianting Zhi, Lihong |
| contents | Jurij Volčič conjectured that a noncommutative polynomial $g$ belongs to the unital $\mathbb{K}$-algebra generated by finitely many noncommutative polynomials if and only if, for matrices of every size, every joint invariant subspace of the evaluations of the generators is also invariant under the evaluation of $g$. In this paper, we establish a homogeneous Nullstellensatz for joint invariant subspaces by proving that this equivalence holds whenever the generators are homogeneous. In contrast, we demonstrate that the statement fails in the general case, thereby settling the conjecture completely. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22233 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Homogeneous Nullstellensatz for Joint Invariant Subspaces Yan, Sizhuo Yang, Jianting Zhi, Lihong Rings and Algebras Operator Algebras Jurij Volčič conjectured that a noncommutative polynomial $g$ belongs to the unital $\mathbb{K}$-algebra generated by finitely many noncommutative polynomials if and only if, for matrices of every size, every joint invariant subspace of the evaluations of the generators is also invariant under the evaluation of $g$. In this paper, we establish a homogeneous Nullstellensatz for joint invariant subspaces by proving that this equivalence holds whenever the generators are homogeneous. In contrast, we demonstrate that the statement fails in the general case, thereby settling the conjecture completely. |
| title | A Homogeneous Nullstellensatz for Joint Invariant Subspaces |
| topic | Rings and Algebras Operator Algebras |
| url | https://arxiv.org/abs/2602.22233 |