Stochastic realisers of non-degenerate full-degree Type II reduced Ito polynomials
Abstract
Let fα(x)=(xq−(1−α))d−αd xz, where 0<α<1, q≥ 2, d≥ 2, 1≤ z≤ 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 fα. The support data are a composition a0+⋯+ad−1=z and a cyclic origin ρ∈ℤq. The transfer support in block 0 is an arbitrary nonempty subset of [ρ,ρ+a0]q, while the support in block t≥ 1 is an arbitrary nonempty subset of [ρ−∑j=1t(aj+1),ρ−∑j=1t(aj+1)+at]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 Θn−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.
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BibTeX
@misc{verbeken2026stochastic,
title = {{Stochastic realisers of non-degenerate full-degree Type II reduced Ito polynomials}},
author = {Brecht F. Verbeken and Vincent Ginis},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26055},
eprint = {2609.26055},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26055},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic realisers of non-degenerate full-degree Type II reduced Ito polynomials
AU - Brecht F. Verbeken
AU - Vincent Ginis
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26055
UR - https://arxiv.org/abs/2609.26055
AB - Let f_α(x)=(x^q−(1−α))^d−α^d x^z, where 0<α<1, q≥ 2, d≥ 2, 1≤ z≤ 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+⋯+a_d−1=z and a cyclic origin ρ∈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≥ 1 is an arbitrary nonempty subset of [ρ−∑_j=1^t(a_j+1),ρ−∑_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.
ER -