Unboundedness of Markov complexity of monomial curves in [formula omitted] for n ≥ 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 A3 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∈N such that m(C)≤d for all monomial curves C in A4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in An, where n≥4.