Hilbert series for ideals generated by generic forms
type: Item
instance of: scholarly article
title: Hilbert series for ideals generated by generic forms
zbMATH Open document ID: 0811.13006
author: Ralf Fröberg
author: Joachim Hollman
published in: Journal of Symbolic Computation
publication date: 1995-05-01T00:00:00Z
review text: A homogeneous ideal \(I = (f_ 1, \dots, f_ g)\) in the polynomial ring \(A = \mathbb{C} [x_ 1, \dots, x_ n]\) is said to be of type \((n,d_ 1, \dots, d_ g)\) if \(\deg f_ i = d_ i\) for each \(i\). The aim is to see how large an ideal \(I\) of given type can be, as measured by the Hilbert series \(\text{Hilb}_{A/I} (t) = \sum \dim_ \mathbb{C} (A/I)_ it^ i\). -- \textit{R. Fröberg} [Math. Scand. 56, 117-144 (1985; Zbl 0582.13007)] found a lower bound for \(\text{Hilb}_{A/I} (t)\) in the lexicographic order for series in \(t\). For suitably ``generic'' ideals \(I\), it is conjectured that \(\text{Hilb}_{A/I} (t)\) is equal to this lower bound. Since generic ideals give the smallest Hilbert series, the production of one ideal, of a given type, which has the conjectured Hilbert series will prove the conjecture for that type. Using an adaptation of Bayer and Stillman's (1990) computer algebra system MACAULAY, the conjecture is proved for the ideal types given by \((n \leq 11\), \(d_ i = 2)\) and \((n \leq 8\), \(d_ i = 3)\). The authors also investigate the extent to which ideals generated by powers of generic linear forms satisfy the conjectured equality.
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 638322
zbMATH Keywords: polynomial ideal
zbMATH Keywords: Hilbert series
zbMATH Keywords: generic ideals
zbMATH Keywords: computer algebra
reviewed by: C. P. L. Rhodes
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: jsco.1994.1008
OpenAlex ID: W2086539677
DOI: 10.1006/JSCO.1994.1008
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description: scientific article; zbMATH DE number 638322