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| Paper IPM / M / 17151 |
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| Abstract: | |
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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?æ , xn]. The k-th squarefree power I[k]
of I is the ideal of S generated by those squarefree monomials u1 â?ï 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?ÃÂ¥ 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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