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Inmathematics themonomial basis of apolynomial ring is itsbasis (as avector space orfree module over thefield orring ofcoefficients) that consists of allmonomials. The monomials form a basis because everypolynomial may be uniquely written as a finitelinear combination of monomials (this is an immediate consequence of the definition of a polynomial).

Thepolynomial ringK[x] ofunivariate polynomials over a fieldK is aK-vector space, which has$${\displaystyle 1,x,x^2,x^3,\dots }$$as an (infinite) basis. More generally, ifK is aring thenK[x] is afree module which has the same basis.

The polynomials ofdegree at mostd form also a vector space (or a free module in the case of a ring of coefficients), which has$${\displaystyle \{1,x,x^2,\dots ,x^{d-1},x^d\}}$$ as a basis.

Thecanonical form of a polynomial is its expression on this basis:$${\displaystyle a_0+a_{1}x+a_2{x}^2+\cdots +a_d{x}^d,}$$or, using the shortersigma notation:$${\displaystyle \sum _{i=0}^d{a}_i{x}^i.}$$

The monomial basis is naturallytotally ordered, either by increasing degreesor by decreasing degrees$${\displaystyle 1>x>x^2>\cdots .}$$

In the case of several indeterminates${\displaystyle x_1,\dots ,x_n,}$ amonomial is a product$${\displaystyle x_1^{d_1}{x}_2^{d_2}\cdots x_n^{d_n},}$$where the${\displaystyle d_i}$ are non-negativeintegers. 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${\displaystyle x_1,\dots ,x_n}$ 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 themonomial basis.

Thehomogeneous polynomials of degree${\displaystyle d}$ form asubspace which has the monomials of degree${\displaystyle d=d_1+\cdots +d_n}$ as a basis. Thedimension of this subspace is the number of monomials of degree${\displaystyle d}$, which is$${\displaystyle (\genfrac{}{}{0ex}{}{d+n-1}{d})=\frac{n(n+1)\cdots (n+d-1)}{d!},}$$where$(\genfrac{}{}{0ex}{}{d+n-1}{d})$ is abinomial 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$${\displaystyle (\genfrac{}{}{0ex}{}{d+n}{d})=(\genfrac{}{}{0ex}{}{d+n}{n})=\frac{(d+1)\cdots (d+n)}{n!}.}$$

In contrast to the univariate case, there is no naturaltotal order of the monomial basis in the multivariate case. For problems which require choosing a total order, such asGröbner basis computations, one generally chooses anadmissiblemonomial order – that is, a total order on the set of monomials such thatand$${\displaystyle 1\le m}$$ for every monomial${\displaystyle m,n,q.}$

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

• Algebra
• Polynomials

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