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Paper   IPM / M / 17151
School of Mathematics
  Title:   The normalized depth function of squarefree powers
  Author(s):  Sara Saeidi madan (Joint with E. Nursel, J. Herzog, and T. Hibi)
  Status:   To Appear
  Journal: Collect. Math.
  Supported by:  IPM
The depth of squarefree powers of a squarefree monomial ideal is introduced. Let I be a squarefree monomial ideal of the polynomial ring S = K[x1, â?¦ , xn]. The k-th squarefree power I[k] of I is the ideal of S generated by those squarefree monomials u1 â?¯ uk with each ui â?? G(I), where G(I) is the unique minimal system of monomial generators of I. Let dk denote the minimum degree of monomials belonging to G(I[k]). One has depth(Sâ??I[k]) â?¥ dk â?? 1. Setting gI(k) = depth(Sâ??I[k])â??(dk â?? 1), one calls gI(k) the normalized depth function of I. The computational experience strongly invites us to propose the conjecture that the normalized depth function is nonincreasing. In the present paper, especially the normalized depth function of the edge ideal of a fnite simple graph is deeply studied.

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