Sharp upper bounds for the Betti numbers of a given Hilbert polynomial (Q380351)

Let \(S = k[x_1,\dots,x_n]\) be a polynomial ring over a field \(k\). Given a Hilbert function \(f(t)\) of some graded ideal \(I\) of \(S\), \textit{F. S. Macaulay} [Proceedings L. M. S. (2) 26, 531--555 (1927; JFM 53.0104.01)] showed that there exists the unique lex ideal \(L\) with the same Hilbert function \(f(t)\). Moreover, \textit{A. M. Bigatti} [Commun. Algebra 21, No. 7, 2317--2334 (1993; Zbl 0817.13007)], \textit{H. A. Hulett} [Commun. Algebra 21, No. 7, 2335--2350 (1993; Zbl 0817.13006)] and \textit{K. Pardue} [Ill. J. Math. 40, No. 4, 564--585 (1996; Zbl 0903.13004)] proved that the Betti numbers of \(L\) are the upper bounds for the Betti numbers of graded ideals having the same Hilbert function \(f(t)\). Now given a Hilbert polynomial \(p(t)\) of some graded ideal \(I\), the authors in the paper under review considered the question of bounding the Betti numbers of saturated graded ideals with the same Hilbert polynomial \(p(t)\). By the result of Bigatti, Hulett and Pardue, one can restrict oneself to the class of saturated lex ideals with the same Hilbert polynomial \(p(t)\). The later class of ideals was considered and classified by \textit{S. Murai} and \textit{T. Hibi} [Proc. Am. Math. Soc. 136, No. 5, 1533--1538 (2008; Zbl 1148.13006)] and \textit{S. Iyengar} and \textit{K. Pardue} [J. Reine Angew. Math. 512, 27--48 (1999; Zbl 0927.13019)]. Using these results, the authors reduced the question to a purely combinatorial question about the ladder sets (see Lemma 3.8 and Proposition 3.9). They devised combinatorial machineries in sections 4 and 5 to settled the combinatorial problem (Theorem 6.1). Moreover, they gave the construction of the saturated graded ideal \(L\) whose Betti numbers achieve the bounds. Some examples are illustrated in section 7.NEWLINENEWLINEThe result has a local analog, which is a natural generalization of the result of \textit{G. Valla} [Compos. Math. 91, No. 3, 305--319 (1994, Zbl 0815.14033)].