--- _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' ...