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_ix^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 $1=x_1^0x_2^0\cdots x_n^0$ 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