MonomialOrder

Q: An option for specifying the MonomialOrder in GroebnerBasis is to provide an explicit weight matrix. But I do not understand the syntax for doing this. Are there examples of this available? In addition, I would like to see a weight matrix for some of the standard monomial orderings.

A: Daniel Lichtblau (danl@wolfram.com) answers: What is required for variables is an rational (or integer) matrix of full rank that comprises a well-founded term ordering. This in turn is fulfilled iff the first nonzero entry in each column is positive. Here is an example from a set of problems posed as a challenge for the ISSAC 1997 conference.

For lexicographic ordering, you can use, for example, an identity matrix.

For degree reverse lexicographic ordering, you can use a matrix with first row set to all ones, next row all zeros except last entry negative one, then move the negative one step to the left in successive rows.