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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.,
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.
Besides respecting multiplication, monomial orders are often required to be well-orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders.
In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions:
Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order.
The choice of a total order on the monomials allows sorting the terms of a polynomial. The leading term of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering).
Concretely, let R be any ring of polynomials. Then the set M of the (monic) monomials in R is a basis of R, considered as a vector space over the field of the coefficients. Thus, any nonzero polynomial p in R has a unique expression as a linear combination of monomials, where S is a finite subset of M and the cu are all nonzero. When a monomial order has been chosen, the leading monomial is the largest u in S, the leading coefficient is the corresponding cu, and the leading term is the corresponding cuu. 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.
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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.,
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.
Besides respecting multiplication, monomial orders are often required to be well-orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders.
In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions:
Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order.
The choice of a total order on the monomials allows sorting the terms of a polynomial. The leading term of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering).
Concretely, let R be any ring of polynomials. Then the set M of the (monic) monomials in R is a basis of R, considered as a vector space over the field of the coefficients. Thus, any nonzero polynomial p in R has a unique expression as a linear combination of monomials, where S is a finite subset of M and the cu are all nonzero. When a monomial order has been chosen, the leading monomial is the largest u in S, the leading coefficient is the corresponding cu, and the leading term is the corresponding cuu. 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.