#### Title

An Extension of Buchberger's Criteria for Gröbner Basis Decision

#### Document Type

Article

#### Publication Date

2010

#### Department

Mathematics

#### Abstract

Two fundamental questions in the theory of Gröbner bases are decision (‘Is a basis *G* of a polynomial ideal a Gröbner basis?’) and transformation (‘If it is not, how do we transform it into a Gröbner basis?’) This paper considers the first question. It is well known that *G* is a Gröbner basis if and only if a certain set of polynomials (the *S*-polynomials) satisfy a certain property. In general there are *m*(*m*−1)/2 of these, where *m* is the number of polynomials in *G*, but criteria due to Buchberger and others often allow one to consider a smaller number. This paper presents two original results. The first is a new characterization theorem for Gröbner bases that makes use of a new criterion that extends Buchberger’s criteria. The second is the identification of a class of polynomial systems *G* for which the new criterion has dramatic impact, reducing the worst-case scenario from *m*(*m*−1)/2 *S*-polynomials to *m*−1.

#### Publication Title

LMS Journal of Computation and Mathematics

#### Volume

13

#### First Page

111

#### Last Page

129

#### Recommended Citation

Perry, J. E.
(2010). An Extension of Buchberger's Criteria for Gröbner Basis Decision. *LMS Journal of Computation and Mathematics, 13*, 111-129.

Available at: http://aquila.usm.edu/fac_pubs/8050