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Paper   IPM / M / 12193
School of Mathematics
  Title:   Stanley depth of powers of the edge ideal of a forest
1.  M. R. Pournaki
2.  S. Yassemi (Joint with S. A. Seyed Fakhari)
  Status:   Published
  Journal: Proc. Amer. Math. Soc.
  Vol.:  141
  Year:  2013
  Pages:   3327-3336
  Supported by:  IPM
Let \mathbbK be a field and S=\mathbbK[x1,...,xn] be the polynomial ring in n variables over the field \mathbbK. Let G be a forest with p connected components G1,…,Gp and let I=I(G) be its edge ideal in S. Suppose that di is the diameter of Gi, 1 ≤ ip, and consider d = max{di | 1 ≤ ip}. Morey has shown that for every t ≥ 1, the quantity max{⎡[(dt+2)/3]⎤+p−1,p} is a lower bound for depth(S/It). In this paper, we show that for every t ≥ 1, the mentioned quantity is also a lower bound for sdepth(S/It). By combining this inequality with Burch's inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.

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