POWERS OF FACET IDEALS OF SIMPLICIAL TREES

Abstract In 2004, Herzog, Hibi, and Zheng proved that a quadratic monomial ideal has a linear resolution if and only if all its powers have a linear resolution. We study a generalization of this result for square-free monomial ideals arising from facet ideals of a simplicial tree. We give a complete characterization of simplicial trees for which all powers of their facet ideal have a linear resolution. We compute the regularity of t -path ideals of rooted trees. In addition, we study the regularity of powers of t -path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $\Delta $ is a d -dimensional simplicial tree connected in codimension one, then reg $(I(\Delta )^s) \leq (d+1)(s-1)~+$ reg $(I(\Delta ))$ for all $s \geq 1$ . We prove this conjecture for some special classes of simplicial trees.