In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of the set of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).

One indeterminateEdit

The polynomial ring K[x] of the univariate polynomial over a field K is a K-vector space, which has


as an (infinite) basis. More generally, if K is a ring, K[x] is a free module, which has the same basis.

The polynomials of degree at most d form also a vector space (or a free module in the case of a ring of coefficients), which has


as a basis

The canonical form of a polynomial is its expression on this basis:


or, using the shorter sigma notation:


The monomial basis is naturally totally ordered, either by increasing degrees


or by decreasing degrees


Several indeterminatesEdit

In the case of several indeterminates   a monomial is a product


where the   are non-negative integers. Note that, as   an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular   is a monomial.

Similar to the case of univariate polynomials, the polynomials in   form a vector space (if the coefficients belong to a field) or a free module (if the coefficients belong to a ring), which has the set of all monomials as a basis, called the monomial basis

The homogeneous polynomials of degree   form a subspace which has the monomials of degree   as a basis. The dimension of this subspace is the number of monomials of degree  , which is


where   denotes a binomial coefficient.

The polynomials of degree at most   form also a subspace, which has the monomials of degree at most   as a basis. The number of these monomials is the dimension of this subspace, equal to


Despite the univariate case, there is no natural total order of the monomial basis. For problem which require to choose a total order, such Gröbner basis computation, one generally chooses an admissible monomial order that is a total order on the set of monomials such that




for every monomials  

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0. For example, a polynomial in  :


See alsoEdit