An Extension of Buchberger's Criteria for Gröbner Basis Decision
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.
LMS Journal of Computation and Mathematics
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