Monomial order

In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., * If u \leq v and w is any other monomial, then uw \leq vw . Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of being a Gröbner basis is always relative to a specific monomial order.

In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., If u \leq v and w is any other monomial, then uw \leq vw . Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of being a Gröbner basis is always relative to a specific monomial order. p = \textstyle\sum_{u \in S} c_u u as a linear combination of monomials, where is a finite subset of and the are all nonzero. When a monomial order has been chosen, the leading monomial is the largest in , the leading coefficient is the corresponding , and the leading term is the corresponding . Head monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial". In this article, a monomial is assumed to not include a coefficient. The defining property of monomial orderings implies that the order of the terms is kept when multiplying a polynomial by a monomial. Also, the leading term of a product of polynomials is the product of the leading terms of the factors. x_1^2 > x_1x_2 > x_1 > x_2^2 > x_2 > 1. For Gröbner basis computations, the lexicographic ordering tends to be the most costly; thus it should be avoided, as far as possible, except for very simple computations. x_1^2 > x_1x_2 > x_2^2 > x_1 > x_2 > 1. Although very natural, this ordering is rarely used: the Gröbner basis for the graded reverse lexicographic order, which follows, is easier to compute and provides the same information on the input set of polynomials.

References

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