Year: 2026
Type: preprint
Abstract: Let $f_\alpha(x)=(x^q-(1-\alpha))^d-\alpha^d x^z$, where $0<\alpha<1$, $q\ge 2$, $d\ge 2$, $1\le z\le q-1$, 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 $f_\alpha$. The support data are a composition $a_0+\cdots+a_{d-1}=z$ and a cyclic origin $\rho\in\mathbb{Z}_q$. The transfer support in block 0 is an arbitrary nonempty subset of $[\rho,\rho+a_0]_q$, while the support in block $t\ge 1$ is an arbitrary nonempty subset of $[\rho-\sum_{j=1}^t(a_j+1),\rho-\sum_{j=1}^t(a_j+1)+a_t]_q$. The horizontal weights at the selected positions are arbitrary elements of $(0,1)$ whose product in every block is $1-\alpha$. The proof first shows that the order-$n$ Type II arc lies strictly outside $\Theta_{n-1}$, 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.
Source: arXiv (Cornell University)
Authors: Verbeken, Brecht F., Ginis, Vincent
Institution: Vrije Universiteit Brussel
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171671580