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 in 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 $\Pi_4$:

$1+x+3x^4$

See also

• Horner's method
• Polynomial sequence
• Newton polynomial
• Lagrange polynomial
• Legendre polynomial
• Bernstein form
• Chebyshev form