# Markov degree of the three-state toric homogeneous Markov chain model

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.

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Journal Article | Published | English

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Author
Haws, David; Martin del Campo Sanchez, AbrahamIST Austria; Takemura, Akimichi; Yoshida, Ruriko
Department
Abstract
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.
Publishing Year
Date Published
2014-03-01
Journal Title
Beitrage zur Algebra und Geometrie
Acknowledgement
Research of Martín del Campo supported in part by NSF Grant DMS-915211.
Volume
55
Issue
1
Page
161 - 188
IST-REx-ID

### Cite this

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
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
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.
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.
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.
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.
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