Resumen de The normalized depth function of squarefree powers

Nursel Erey, Jürgen Herzog , Takayuki Hibi, Sara Saeedi Madani

• 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.