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Powers of facet ideals of simplicial trees

In this article, we study the linearity of the minimal free resolution of powers of facets ideals of simplicial trees. We give a complete characterization of simplicial trees for which (some) power of its facet ideal has a linear resolution. We calculate the regularity of the $t$-path ideal of a perfect rooted tree. We also obtain an upper bound for the regularity of the $t$-path ideal of a rooted tree. We give a procedure to calculate the regularity of powers of facet ideals of simplicial trees. As a consequence of this result, 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 $Δ$ is a $d$-dimensional simplicial tree, then $\reg(I(Δ)^s) \leq (d+1)(s-1)+\reg(I(Δ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees.