Monomial ideals and \(n\)-lists (Q1765966)

Generalizing a construction of \textit{A. V. Geramita}, \textit{T. Harima} and \textit{Y. S. Shin} [Ill. J. Math 45, 1--23 (2001; Zbl 1095.13500)], the author introduces so-called \(n\)-lists: A \(1\)-list is a natural number, and for \(n\geq 1\) an \(n\)-list is a decreasing infinite sequence of \((n- 1)\)-lists, where \(A\geq B\) for two \(n\)-lists \(A= (A_i)\) and \(B= (B_i)\) if \(A_i\geq B_i\) for all \(i\). Next an injective map \(\Phi\) from the set of \(n\)-lists to the set of monomial ideals in the polynomial ring \(k[x_1,\dots, x_n]\) (\(k\) a field) is defined. (\(\Phi\) is not surjective which does'nt affect the applications given by the author. In a final section the author shows how one may generalize the set of \(n\)-lists to make \(\Phi\) surjective.) The map \(\Phi\) serves to characterize the Artinian monomial ideals in terms of \(n\)-lists. Furthermore, it is used to calculate the multiplicities and Hilbert polynomial degrees for the quotients of Borel fixed ideals, and to give a new proof of the result due to Geramita, Harima, and Shin (loc. cit.).