A Note on Multivariate Polynomial Division and Gr\"Obner Bases

Aleksandar T. Lipkovski, Samira Zeada

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Gr\"obner, and to suggest a new heuristic approach to Gr\"obner basis computations.