Markov complexity of monomial curves

Abstract

Let $\mathcal{A}=\{{\mathbf{a}}_1,\dots ,{\mathbf{a}}_n\}\subset {\mathbb{N}}^m$. We give an algebraic characterization of the universal Markov basis of the toric ideal $I_{\mathcal{A}}$. We show that the Markov complexity of $\mathcal{A}=\{n_1,n_2,n_3\}$ is equal to two if $I_{\mathcal{A}}$ is complete intersection and equal to three otherwise, answering a question posed by Santos and Sturmfels. We prove that for any $r\ge 2$ there is a unique minimal Markov basis of ${\mathcal{A}}^{(r)}$. Moreover, we prove that for any integer $l$ there exist integers $n_1,n_2,n_3$ such that the Graver complexity of $\mathcal{A}$ is greater than $l$.

19 pages