Monomial basis

Short description: Basis of polynomials consisting of monomials

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 $$1,x,x^2,x^3,\dots$$ 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 $$1,x,x^2,\dots$$ 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: $$\sum _{i=0}^d{a}_i{x}^i.$$

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 $x_1,\dots ,x_n,$ 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. 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 $x_1,\dots ,x_n$ 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 $$(\genfrac{}{}{0ex}{}{d+n-1}{d})=\frac{n(n+1)\cdots (n+d-1)}{d!},$$ where $(\genfrac{}{}{0ex}{}{d+n-1}{d})$ 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 $$(\genfrac{}{}{0ex}{}{d+n}{d})=(\genfrac{}{}{0ex}{}{d+n}{n})=\frac{(d+1)\cdots (d+n)}{n!}.$$

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 $$m<n⟺mq<nq$$ and $$1\le m$$ for every monomial $m,n,q.$

See also

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