Unboundedness of Markov complexity of monomial curves in ${\mathbb{A}}^n$ for $n\ge 4$

Abstract

Computing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve $C$ in ${\mathbb{A}}^3$ has Markov complexity $m(C)$ two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no $d\in \mathbb{N}$ such that $m(C)\le d$ for all monomial curves $C$ in ${\mathbb{A}}^4$. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in ${\mathbb{A}}^n,n\ge 4$.

arXiv comment12 pages, 1 table

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• Higher Lawrence configurations