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 K[x] of univariate polynomials over a field K is a K-vector space, which has

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

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

$${\displaystyle \{1,x,x^{2},\ldots ,x^{d-1},x^{d}\))$$

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

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

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

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

$${\displaystyle 1<x<x^{2}<\cdots ,}$$

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

Several indeterminates

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

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

${\displaystyle d_{i))$

${\displaystyle x_{i}^{0}=1,}$

${\displaystyle 1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0))$

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

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

${\textstyle {\binom {d+n-1}{d))}$

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

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

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

$${\displaystyle m<n\iff mq<nq}$$

$${\displaystyle 1\leq m}$$

${\displaystyle m,n,q.}$