Stochastic realisers of non-degenerate full-degree Type II reduced Ito polynomials

Abstract

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.