发表机构
School of Mathematics, Nanjing University; Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; University of the Chinese Academy of Sciences; School of Intelligence Science and Technology, Peking University（南京大学数学系; 中国科学院数学与系统科学研究院晨兴数学中心；中国科学院大学; 北京大学智能学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过处理奇数情形并利用次数分解、图论方法及维度扩展论证，完成了Dixon-Pressman关于广义交换子算子通用零化度猜想的证明。
AI 中文摘要
我们研究矩阵代数上广义交换子算子\\[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\\]的通用零化度，其中$s_{k+1}$表示标准多项式。Dixon和Pressman猜想了一个关于$L_{\mathbf{A}}$的通用零化度的显式公式，Brassil和Reichstein在$k$为偶数时证明了该猜想。在本文中，我们解决了$k$为奇数时的剩余情况。我们的证明首先利用次数分解和交错迹形式的图论解释处理维度$n$和$n+1$中的边界情况$k=2n-3$，然后建立从$n$到$n+2$的维度扩展论证。因此，结合Brassil和Reichstein的结果，我们在任意特征为零的域上完成了Dixon-Pressman通用零化度猜想的完整证明。
英文摘要
We study the generic nullity of generalized commutator operators \[L_{\mathbf{A}}(X)=s_{k+1}(A_1,\cdots,A_k,X)\] on matrix algebras, where $s_{k+1}$ denotes the standard polynomial. Dixon and Pressman conjectured an explicit formula for the generic nullity of $L_{\mathbf{A}}$, 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.
Comments21 pages
