Monomial basis

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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[edit]

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

Notes[edit]

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

Examples[edit]

A polynomial in \Pi_4

1+x+3x^4

See also[edit]