Monomial basis

In mathematics the monomial basis of a polynomial ring is its basis that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials.

One indeterminate

The polynomial ring of univariate polynomials over a field is a -vector space, which has
as an 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, which has as a basis.
The canonical form of a polynomial is its expression on this basis:
or, using the shorter sigma notation:
The monomial basis is naturally totally ordered, either by increasing degrees
or by decreasing degrees

Several indeterminates

In the case of several indeterminates a monomial is a product
where the are non-negative integers. As an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular is a monomial.
Similar to the case of univariate polynomials, the polynomials in form a vector space or a free module, which has the set of all monomials as a basis, called the monomial basis.
The homogeneous polynomials of degree form a subspace which has the monomials of degree as a basis. The dimension of this subspace is the number of monomials of degree, which is
where is a binomial coefficient.
The polynomials of degree at most form also a subspace, which has the monomials of degree at most as a basis. The number of these monomials is the dimension of this subspace, equal to
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
and for every monomial

In analysis and numerical applications

The coefficients of a polynomial in a monomial basis represent the local behavior near the origin in the complex plane, and are proportional to the values of the various derivatives of the function there. When the series is the Taylor series of some non-polynomial function, it will converge within an origin-centered disk in the complex plane, whose radius is as large as possible such that the function is analytic inside the disk.
Polynomials in monomial basis are generally poor choices for numerical evaluation away from the origin, and other polynomial bases are much better suited for representing a polynomial over a specific real interval or arbitrary region in the complex plane.