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

TL;DR

This paper proves that the Markov complexity of monomial curves in affine spaces of dimension four or higher is unbounded, highlighting the complexity's unpredictable growth and the difficulty of computing Markov bases in higher dimensions.

Contribution

It establishes that Markov complexity is unbounded for monomial curves in dimensions four and above, extending previous results from three dimensions.

Findings

Markov complexity is bounded for curves in a33.

Unboundedness of Markov complexity in a34 and higher.

Even complete intersection monomial curves in a34 are unbounded.

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 $m(C)\leq d$ 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 $\mathbb{A}^n, n\geq 4$.