Ideals generated by quadrics exhibiting double exponential degrees
type: Item
instance of: scholarly article
title: Ideals generated by quadrics exhibiting double exponential degrees
zbMATH Open document ID: 0924.13022
published in: Journal of Algebra
publication date: 1999-11-07T00:00:00Z
review text: Let \(I=(h_1,\dots,h_s)\) be an ideal in the polynomial ring \(R=K[x_1,\dots,x_n]\) over a field. A measure of the computational cost of algorithms solving the ideal membership problem is obtained by bounding the degrees of coefficients \(g_i\) which espress elements \(h\in I\) as \(h=\sum g_ih_i\). Actually we have bounds which are doubly exponential in the number \(n\) of variables; on the other hand, \textit{E. W. Mayr} and \textit{A. R. Meyer} [Adv. Math. 46, 305-329 (1982; Zbl 0506.03007] constructed examples of elements in some polynomial rings, requiring coefficients whose degrees have a doubly exponential growth; it turns out that known bounds can be possibly refined, but not substantially improved. The examples of Mayr and Meyer (loc. cit.) are generated in degree \(5\). In the present paper, the author modifies these examples, providing ideals generated by quadrics for which the doubly exponential growth occurs.
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68Q25
zbMATH DE Number: 1139776
zbMATH Keywords: polynomial ring
zbMATH Keywords: computational cost of algorithms
zbMATH Keywords: ideal membership problem
zbMATH Keywords: ideals generated by quadrics
author: Jee Koh
reviewed by: Luca Chiantini
MaRDI profile type: Publication
full work available at URL: jabr.1997.7225
OpenAlex ID: W2017089594
cites work: On the complexity of computing syzygies
cites work: A note on upper bounds for ideal-theoretic problems
cites work: The complexity of the word problems for commutative semigroups and polynomial ideals
cites work: Constructions in Algebra
cites work: Fields of large transcendence degree generated by values of elliptic functions
cites work: A new lower bound construction for commutative Thue systems with applications
DOI: 10.1006/JABR.1997.7225
description: scientific article; zbMATH DE number 1139776