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Strong Gröbner bases for polynomials over a principal ideal ring

Published online by Cambridge University Press:  17 April 2009

Graham H. Norton  and
Ana Sǎlǎgean
Graham H. Norton
Affiliation:
Department of Mathematics, University of Queensland, Brisbane, Qld. 4072, Australia e-mail: [email protected] URL's: http://www.maths.uq.edu.au/~ghn
Ana Sǎlǎgean
Affiliation:
Department of computer Science, Loughborough University, Leicestershire LE11 3TU, United Kingdom e-mail: [email protected]://www.lboro.ac.uk/departments/co/personal_pages/salagean.html.
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Abstract

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Gröbner bases have been generalised to polynomials over a commutative ring A in several ways. Here we focus on strong Gröbner bases, also known as D-bases. Several authors have shown that strong Gröbner bases can be effectively constructed over a principal ideal domain. We show that this extends to any principal ideal ring. We characterise Gröbner bases and strong Gröbner bases when A is a principal ideal ring. We also give algorithms for computing Gröbner bases and strong Gröbner bases which generalise known algorithms to principal ideal rings. In particular, we give an algorithm for computing a strong Gröbner basis over a finite-chain ring, for example a Galois ring.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 3 , December 2001 , pp. 505 - 528
Copyright
Copyright © Australian Mathematical Society 2001

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