Monomial basis

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

$${\displaystyle 1,x,x^{2},x^{3},\ldots }$$

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

$${\displaystyle 1,x,x^{2},\ldots }$$

as a basis.

The canonical form of a polynomial is its expression on this basis:

$${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{d}x^{d},}$$

or, using the shorter sigma notation:

$${\displaystyle \sum _{i=0}^{d}a_{i}x^{i}.}$$

The monomial basis is naturally totally ordered, either by increasing degrees

$${\displaystyle 1<x<x^{2}<\cdots ,}$$

or by decreasing degrees

$${\displaystyle 1>x>x^{2}>\cdots .}$$

In the case of several indeterminates ${\displaystyle x_{1},\ldots ,x_{n},}$ a monomial is a product

$${\displaystyle x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},}$$

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

$${\displaystyle {\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},}$$

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

$${\displaystyle {\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\cdots (d+n)}{n!}}.}$$

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

$${\displaystyle m<n\iff mq<nq}$$

and

$${\displaystyle 1\leq m}$$

for every monomial ${\displaystyle m,n,q.}$

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