Monomial basis


In mathematics the monomial basis of a polynomial ring is its basis 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.

One indeterminate

The polynomial ring of the univariate polynomial over a field is a -vector space, which has
as an basis. More generally, if is a ring, is a free module, which has the same basis.
The polynomials of degree at most form also a vector space, 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

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 or a free module, 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 :