Borel sets and sectional matrices
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.
scientific article; zbMATH DE number 1160052