Monomial 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 K[x] of univariate polynomials over a field K is a K-vector space, which has

as an (infinite) basis. More generally, if K is a ring then 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

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 ${\displaystyle x_{1},\ldots ,x_{n},}$ a monomial is a product

where the ${\displaystyle d_{i}}$ are non-negative integers. As ${\displaystyle x_{i}^{0}=1,}$ an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular ${\displaystyle 1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}}$ is a monomial.

Similar to the case of univariate polynomials, the polynomials in ${\displaystyle x_{1},\ldots ,x_{n}}$ 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 ${\displaystyle d}$ form a subspace which has the monomials of degree ${\displaystyle d=d_{1}+\cdots +d_{n}}$ as a basis. The dimension of this subspace is the number of monomials of degree ${\displaystyle d}$, which is

where ${\textstyle {\binom {d+n-1}{d}}}$ is a binomial coefficient.

The polynomials of degree at most ${\displaystyle d}$ form also a subspace, which has the monomials of degree at most ${\displaystyle d}$ 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 ${\displaystyle m,n,q.}$

See also

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