Monomial basis

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 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.
The polynomial ring of univariate polynomials over a field is a -vector space, which has 
as an (infinite) basis. More generally, if is a ring then is a free module which has the same basis.
The polynomials of degree at most 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.
In the case of several indeterminates formula_7 a monomial is a product
where the formula_9 are non-negative integers. As formula_10 an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular
formula_11
is a monomial.
Similar to the case of univariate polynomials, the polynomials in formula_12 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 formula_13 form a subspace which has the monomials of degree formula_14 as a basis. The dimension of this subspace is the number of monomials of degree formula_13, which is 
where formula_17 is a binomial coefficient.
The polynomials of degree at most formula_13 form also a subspace, which has the monomials of degree at most formula_13 as a basis. The number of these monomials is the dimension of this subspace, equal to
In contrast to the univariate case, there is no natural total order of the monomial basis in the multivariate case. For problems which require choosing a total order, such as Gröbner basis computations, one generally chooses an "admissible" monomial order – that is, a total order on the set of monomials such that
and
for every monomial formula_23