Monomial basis

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

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<x<x^{2}<\cdots ,

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<n\Leftrightarrow mq<nq

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

Monomial basis

One indeterminate

Several indeterminates

See also