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Generic Nullity of Generalized Commutators

  • Chen Lin
  • Chenhao Tang
  • Enhan Zhao
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Abstract

We study the generic nullity of generalized commutator operators LA(X)=sk+1(A1,⋯,Ak,X) on matrix algebras, where sk+1 denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of LA, and Brassil and Reichstein proved the conjecture when k is even. In this paper, we settle the remaining case where k is odd. Our proof first treats the boundary cases k=2n−3 in dimensions n and n+1 using degree decompositions and graph-theoretic interpretations of alternating trace forms, and then establishes a dimension-extension argument from n to n+2. Consequently, together with the result of Brassil and Reichstein, we obtain a complete proof of the Dixon-Pressman generic nullity conjecture over any field of characteristic zero.

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Cite

BibTeXRIS
BibTeX
@misc{lin2026generic,
  title = {{Generic Nullity of Generalized Commutators}},
  author = {Chen Lin and Chenhao Tang and Enhan Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06339},
  eprint = {2609.06339},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06339},
}
RIS
TY  - UNPB
TI  - Generic Nullity of Generalized Commutators
AU  - Chen Lin
AU  - Chenhao Tang
AU  - Enhan Zhao
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06339
UR  - https://arxiv.org/abs/2609.06339
AB  - We study the generic nullity of generalized commutator operators L_A(X)=s_k+1(A_1,⋯,A_k,X) on matrix algebras, where s_k+1 denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of L_A, and Brassil and Reichstein proved the conjecture when k is even. In this paper, we settle the remaining case where k is odd. Our proof first treats the boundary cases k=2n−3 in dimensions n and n+1 using degree decompositions and graph-theoretic interpretations of alternating trace forms, and then establishes a dimension-extension argument from n to n+2. Consequently, together with the result of Brassil and Reichstein, we obtain a complete proof of the Dixon-Pressman generic nullity conjecture over any field of characteristic zero.
ER  -