# 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 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

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

$1,x,x^{2},x^{3},\ldots$

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

$1,x,x^{2},\ldots$

as a basis

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

$a_{0}+a_{1}x+a_{2}x^{2}+\ldots +a_{d}x^{d},$

or, using the shorter sigma notation:

$\sum _{i=0}^{d}a_{i}x^{i}.$

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

$1

or by decreasing degrees

$1>x>x^{2}>\cdots .$

## Several indeterminates

In the case of several indeterminates $x_{1},\ldots ,x_{n},$  a monomial is a product

$x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},$

where the $d_{i}$  are non-negative integers. Note that, as $x_{i}^{0}=1,$  an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular $1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}$  is a monomial.

Similar to the case of univariate polynomials, the polynomials in $x_{1},\ldots ,x_{n}$  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 $d$  form a subspace which has the monomials of degree $d=d_{1}+\cdots +d_{n}$  as a basis. The dimension of this subspace is the number of monomials of degree $d$ , which is

${\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},$

where ${\binom {d+n-1}{d}}$  denotes a binomial coefficient.

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

${\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\cdots (d+n)}{n!}}.$

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

$m

and

$1\leq m$

for every monomials $m,n,q.$

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0. For example, a polynomial in $\Pi _{4}$ :

$1+x+3x^{4}$