Title:  Conditions for the toric homogenous Markov Chain models to have square-free quadratic Groebner basis

Abstract:  Discrete time Markov chains are often used in statistical models to fit the observed data from a random physical process. Sometimes, in order to simplify the model, it is convenient to consider time-homogeneous Markov chains, where the transition probabilities do not depend on the time.  While under the time-homogeneous Markov chain model it is assumed that the row sums of the transition probabilities are equal to one, under the the toric homogeneous Markov chain (THMC) model the parameters are free and the row sums of the transition probabilities are not restricted.

In this talk we consider a Markov basis and a Groebner basis for the toric ideal associate with the design matrix (configuration) defined by THMC model with the state space with $m$ states where $m \geq 2$ and we study when THMC with $m$ states have a square-free quadratic Groebner basis.  One such example is the embedded discrete Markov chain for the Kimura three parameter model. This is joint work with Abraham Martin del Campo and Akimichi Takemura.