Monomial basis

In mathematics the monomial basis of a polynomial ring is its basis (as vector space or free module over the field or ring of coefficients) that consists in the set of all monomials. In fact, a polynomial may be uniquely written as a linear combination of monomials.

Univariate polynomials expressed on the monomial basis can be evaluated efficiently using Horner's method.

Definition

The monomial basis for the vector space $\Pi _{n}$ of polynomials with degree n is the polynomial sequence of monomials

1,x,x^{2},.\ldots ,x^{n}

The monomial form of a polynomial $p\in \Pi _{n}$ is a linear combination of monomials

a_{0}1+a_{1}x+a_{2}x^{2}+\ldots +a_{n}x^{n}

alternatively the shorter sigma notation can be used

p=\sum _{\nu =0}^{n}a_{\nu }x^{\nu }

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0.

A polynomial in $\Pi _{4}$

1+x+3x^{4}