Abstract
The depth of squarefree powers of a squarefree monomial ideal is introduced. Let I be a squarefree monomial ideal of the polynomial ring \(S=K[x_1,\ldots ,x_n]\). The k-th squarefree power \(I^{[k]}\) of I is the ideal of S generated by those squarefree monomials \(u_1\cdots u_k\) with each \(u_i\in G(I)\), where G(I) is the unique minimal system of monomial generators of I. Let \(d_k\) denote the minimum degree of monomials belonging to \(G(I^{[k]})\). One has \({\text {depth}}(S/I^{[k]}) \ge d_k -1\). Setting \(g_I(k) = {\text {depth}}(S/I^{[k]}) - (d_k - 1)\), one calls \(g_I(k)\) 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 finite simple graph is deeply studied.
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Acknowledgements
Jürgen Herzog and Sara Saeedi Madani was supported by TÜBİTAK (2221-Fellowships for Visiting Scientists and Scientists on Sabbatical Leave) to visit Nursel Erey at Gebze Technical University. Takayuki Hibi was partially supported by JSPS KAKENHI 19H00637. Sara Saeedi Madani was in part supported by a grant from IPM (No. 1401130112). We thank the referee for his/her careful reading and useful suggestions.
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Erey, N., Herzog, J., Hibi, T. et al. The normalized depth function of squarefree powers. Collect. Math. 75, 409–423 (2024). https://doi.org/10.1007/s13348-023-00392-x
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DOI: https://doi.org/10.1007/s13348-023-00392-x