非退化全次数II型约化Ito多项式的随机实现

中文摘要 AI 辅助

设$f_\alpha(x)=(x^q-(1-\alpha))^d-\alpha^d x^z$，其中$0<\alpha<1$，$q\ge 2$，$d\ge 2$，$1\le z\le q-1$，且$\gcd(q,z)=1$。这些是$n=qd$阶的非退化全次数II型约化Ito多项式。我们给出了特征多项式为$f_\alpha$的随机矩阵在置换相似意义下的完整参数化。支撑数据是一个组合$a_0+\cdots+a_{d-1}=z$和一个循环原点$\rho\in\mathbb{Z}_q$。块0中的转移支撑是$[\rho,\rho+a_0]_q$的任意非空子集，而块$t\ge 1$中的支撑是$[\rho-\sum_{j=1}^t(a_j+1),\rho-\sum_{j=1}^t(a_j+1)+a_t]_q$的任意非空子集。所选位置的水平权重是$(0,1)$中的任意元素，其在每个块中的乘积为$1-\alpha$。证明首先表明$n$阶II型弧严格位于$\Theta_{n-1}$之外，这允许使用Dmitriev--Dynkin/Kirkland--Smigoc两位移动约化。然后一个圆形分离定理对可能的支撑进行分类，Coates公式提供特征多项式。这解决了非稀疏全次数II型特征多项式实现问题。

英文摘要

Let $f_α(x)=(x^q-(1-α))^d-α^d x^z$, where $0<α<1$, $q\ge 2$, $d\ge 2$, $1\le z\le q-1$, and $\gcd(q,z)=1$. These are the non-degenerate full-degree Type II reduced Ito polynomials of order $n=qd$. We give a complete parametrisation, up to permutation similarity, of the stochastic matrices whose characteristic polynomial is $f_α$. The support data are a composition $a_0+\cdots+a_{d-1}=z$ and a cyclic origin $ρ\in\mathbb{Z}_q$. The transfer support in block 0 is an arbitrary nonempty subset of $[ρ,ρ+a_0]_q$, while the support in block $t\ge 1$ is an arbitrary nonempty subset of $[ρ-\sum_{j=1}^t(a_j+1),ρ-\sum_{j=1}^t(a_j+1)+a_t]_q$. The horizontal weights at the selected positions are arbitrary elements of $(0,1)$ whose product in every block is $1-α$. The proof first shows that the order-$n$ Type II arc lies strictly outside $Θ_{n-1}$, which permits the Dmitriev--Dynkin/Kirkland--Smigoc two-shift reduction. A circular separation theorem then classifies the possible supports, and Coates' formula supplies the characteristic polynomial. This resolves the non-sparse full-degree Type II characteristic-polynomial realiser problem.

发表机构

• Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）
• imec-SMIT, Vrije Universiteit Brussel（imec-SMIT，布鲁塞尔自由大学）
• Harvard University（哈佛大学）

机构由 AI 辅助整理，请以论文原文为准。

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22 pages, 2 figures