Monomial Order - Related Notions

Related Notions

• An elimination order guarantees that a monomial involving any of a set of indeterminates will always be greater than a monomial not involving any of them.

• A product order is the easier example of an elimination order. It consists in combining monomial orders on disjoint sets of indeterminates into a monomial order on their union. It simply compares the exponents of the indeterminates in the first set using the first monomial order, then breaks ties using the other monomial ordering on the indeterminates of the second set. This method obviously generalizes to any disjoint union of sets of intertermines; the lexicographic order can be so obtained from the singleton sets {x₁}, {x₂}, {x₃}, ... (with the unique monomial ordering for each singleton).

When using monomial orderings to define Gröbner bases, different orders can lead to different results. For example, graded reverse lexicographic order has a reputation for producing relatively small Gröbner bases, while elimination orders can be used with the same algorithms to solve systems of polynomial equations by eliminating variables.