Polynomial 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 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 of univariate polynomials over a field is a -vector space, which has
$1, x, x^2, x^3, \ldots$
as an (infinite) basis. More generally, if is a ring then is a free module which has the same basis.

The polynomials of degree at most form also a vector space (or a free module in the case of a ring of coefficients), which has $\$ as a basis.

The canonical form of a polynomial is its expression on this basis:
$a_0 + a_1 x + a_2 x^2 + \dots + a_d x^d,$
or, using the shorter sigma notation:
$\sum_^d a_ix^i.$

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
$x_1^x_2^\cdots x_n^,$
where the $d_i$ are non-negative integers. 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
$\binom = \frac,$
where $\binom$ is 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
$\binom= \binom=\frac{n!}.$

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
m
and $1 \leq m$ for every monomial $m, n, q.$

See also

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

Algebra
Polynomials