# 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

The **monomial basis** for the vector space of polynomials with degree *n* is the polynomial sequence of monomials

The **monomial form** of a polynomial is a linear combination of monomials

alternatively the shorter sigma notation can be used

## Notes

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