Monomial basis

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In mathematics the monomial basis of a polynomial ring is its basis (as vector space or free module over the field or ring of coefficients) that consists in 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 indeterminate[edit]

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 indeterminates[edit]

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

and

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 also[edit]