---
_id: '2178'
abstract:
- lang: eng
text: We consider the three-state toric homogeneous Markov chain model (THMC) without
loops and initial parameters. At time T, the size of the design matrix is 6 ×
3 · 2T-1 and the convex hull of its columns is the model polytope. We study the
behavior of this polytope for T ≥ 3 and we show that it is defined by 24 facets
for all T ≥ 5. Moreover, we give a complete description of these facets. From
this, we deduce that the toric ideal associated with the design matrix is generated
by binomials of degree at most 6. Our proof is based on a result due to Sturmfels,
who gave a bound on the degree of the generators of a toric ideal, provided the
normality of the corresponding toric variety. In our setting, we established the
normality of the toric variety associated to the THMC model by studying the geometric
properties of the model polytope.
acknowledgement: Research of Martín del Campo supported in part by NSF Grant DMS-915211.
author:
- first_name: David
full_name: Haws, David
last_name: Haws
- first_name: Abraham
full_name: Martin Del Campo Sanchez, Abraham
id: 4CF47F6A-F248-11E8-B48F-1D18A9856A87
last_name: Martin Del Campo Sanchez
- first_name: Akimichi
full_name: Takemura, Akimichi
last_name: Takemura
- first_name: Ruriko
full_name: Yoshida, Ruriko
last_name: Yoshida
citation:
ama: Haws D, Martin del Campo Sanchez A, Takemura A, Yoshida R. Markov degree of
the three-state toric homogeneous Markov chain model. Beitrage zur Algebra
und Geometrie. 2014;55(1):161-188. doi:10.1007/s13366-013-0178-y
apa: Haws, D., Martin del Campo Sanchez, A., Takemura, A., & Yoshida, R. (2014).
Markov degree of the three-state toric homogeneous Markov chain model. Beitrage
Zur Algebra Und Geometrie. Springer. https://doi.org/10.1007/s13366-013-0178-y
chicago: Haws, David, Abraham Martin del Campo Sanchez, Akimichi Takemura, and Ruriko
Yoshida. “Markov Degree of the Three-State Toric Homogeneous Markov Chain Model.”
Beitrage Zur Algebra Und Geometrie. Springer, 2014. https://doi.org/10.1007/s13366-013-0178-y.
ieee: D. Haws, A. Martin del Campo Sanchez, A. Takemura, and R. Yoshida, “Markov
degree of the three-state toric homogeneous Markov chain model,” Beitrage zur
Algebra und Geometrie, vol. 55, no. 1. Springer, pp. 161–188, 2014.
ista: Haws D, Martin del Campo Sanchez A, Takemura A, Yoshida R. 2014. Markov degree
of the three-state toric homogeneous Markov chain model. Beitrage zur Algebra
und Geometrie. 55(1), 161–188.
mla: Haws, David, et al. “Markov Degree of the Three-State Toric Homogeneous Markov
Chain Model.” Beitrage Zur Algebra Und Geometrie, vol. 55, no. 1, Springer,
2014, pp. 161–88, doi:10.1007/s13366-013-0178-y.
short: D. Haws, A. Martin del Campo Sanchez, A. Takemura, R. Yoshida, Beitrage Zur
Algebra Und Geometrie 55 (2014) 161–188.
date_created: 2018-12-11T11:56:10Z
date_published: 2014-03-01T00:00:00Z
date_updated: 2021-01-12T06:55:48Z
day: '01'
department:
- _id: CaUh
doi: 10.1007/s13366-013-0178-y
intvolume: ' 55'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1204.3070
month: '03'
oa: 1
oa_version: Submitted Version
page: 161 - 188
publication: Beitrage zur Algebra und Geometrie
publication_status: published
publisher: Springer
publist_id: '4804'
quality_controlled: '1'
scopus_import: 1
status: public
title: Markov degree of the three-state toric homogeneous Markov chain model
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 55
year: '2014'
...