Regularity of symbolic powers and Arboricity of matroids

Let $\mathrm{\Delta }$ 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 $I_{\mathrm{\Delta }}$ 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 $M^{\ast }$.