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

  • David Haws
  • Abraham Martín del Campo
  • Akimichi Takemura
  • Ruriko Yoshida
Original Paper

DOI: 10.1007/s13366-013-0178-y

Cite this article as:
Haws, D., Martín del Campo, A., Takemura, A. et al. Beitr Algebra Geom (2014) 55: 161. doi:10.1007/s13366-013-0178-y


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 \(6 \times 3\cdot 2^{T-1}\) and the convex hull of its columns is the model polytope. We study the behavior of this polytope for \(T\ge 3\) and we show that it is defined by \(24\) facets for all \(T\ge 5\). 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.


Toric ideals Toric homogeneous Markov chains Polyhedron Semigroups 

Mathematics Subject Classification


Copyright information

© The Managing Editors 2013

Authors and Affiliations

  • David Haws
    • 1
  • Abraham Martín del Campo
    • 2
  • Akimichi Takemura
    • 3
  • Ruriko Yoshida
    • 4
  1. 1.IBM, Watson in Yorktown HeightsNew YorkUSA
  2. 2.IST AustriaKlosterneuburgAustria
  3. 3.University of TokyoTokyoJapan
  4. 4.University of KentuckyLexingtonUSA

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