Borel sets and sectional matrices
type: Item
instance of: scholarly article
title: Borel sets and sectional matrices
zbMATH Open document ID: 0918.13009
DOI: 10.1007/BF02558475
published in: Annals of Combinatorics
publication date: 1999-08-19T00:00:00Z
review text: Let \(I\) be a homogeneous ideal of the ring of polynomials \(k[x_1, x_2,\dots, x_n]\) over a field \(k\) of zero characteristic. The authors define the sectional matrix of \(I\) by its elements \[ M_I(i,d)= H_{(I+(L_1,L_2,\dots, L_{n-i}))/ (L_1, L_2,\dots, L_{n-i})} (d), \] where \(L_1, L_2,\dots, L_{n-i}\) are general linear forms and the expression on the right side is the value of the Hilbert function (of \(d\)) of the corresponding module. Properties of the sectional matrix, embodying the inequalities of \textit{F. S. Macaulay} [Proc. Lond. Math. Soc. (2) 26, 531-555 (1927; JFM 53.0104.01)] and \textit{M. Green} [in: Algebraic curves and projective geometry, Proc. Conf. Trento 1988, Lect. Notes Math. 1389, 76-86 (1989; Zbl 0717.14002)], are indicated, as well as the persistence theorem: \[ P(d) \Rightarrow (t>d\Rightarrow P(t)), \] where \(P(t)\) is the predicate, corresponding to the equality: \(M_I(i,t+1)= \sum_{j=1}^i M_I(j,t)\) and the generators of \(I\) have degrees \(\leq d\). As corollaries, the corresponding theorem of \textit{G. Gotzmann} [Math. Z. 158, 61-70 (1978; Zbl 0352.13009)] and a result of \textit{M. Miller} and \textit{R. H. Villarreal} [Proc. Am. Math. Soc. 124, No. 2, 377-382 (1996; Zbl 0846.13010)] are obtained. The used method of proof consists in reduction to the case of so-called monomial Borel ideals by means of Galligo's theorem [\textit{A. Galligo}, in: Fonctions de plusieurs variables complexes, Sémin. F. Norguet, Lect. Notes Math. 409, 543-579 (1974; Zbl 0297.32003); \textit{D. Bayer} and \textit{M. Stillman}, Duke Math. J. 55, 321-328 (1987; Zbl 0638.06003)] and their combinatorial investigation.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 1160052
zbMATH Keywords: combinatorial enumeration problems
zbMATH Keywords: Hilbert function
zbMATH Keywords: monomial Borel ideals
zbMATH Keywords: JFM 53.0104.01
author: Anna M. Bigatti
author: Lorenzo Robbiano
MaRDI profile type: Publication
cites work: Gotzmann theorems for exterior algebras and combinatorics
cites work: Q5563439
cites work: Upper bounds for the betti numbers of a given hilbert function
cites work: A criterion for detecting m-regularity
cites work: A theorem on refining division orders by the reverse lexicographic order
cites work: Minimal resolutions of some monomial ideals
cites work: Extremal properties of Hilbert functions
cites work: Maximum betti numbers of homogeneous ideals with a given hilbert function
cites work: Q4470473
cites work: A note on generators of least degree in Gorenstein ideals
description: scientific article; zbMATH DE number 1160052