Regularity of symbolic powers and Arboricity of matroids

Abstract

Let $\Delta$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_{\Delta }$ in terms of combinatorial data of the matroid $M$. In order to do that, we provide a sharp bound between the arboricity of $M$ and the circumference of its dual $M^{\ast }$.