monomial basis

The polynomial ring {{math|K[x]}} of univariate polynomials over a field {{math|K}} is a {{math|K}}-vector space, which has

$$1,\; x,\; x^2,\; x^3,\; \backslash ldots$$

as an (infinite) basis. More generally, if {{math|K}} is a ring then {{math|K[x]}} is a free module which has the same basis.

The polynomials of degree at most {{math|d}} form also a vector space (or a free module in the case of a ring of coefficients), which has $$\backslash \{\; 1,\; x,\; x^2,\; \backslash ldots,\; x^\{d-1\},\; x^d\; \backslash \}$$ as a basis.

The canonical form of a polynomial is its expression on this basis:

$$a\_0\; +\; a\_1\; x\; +\; a\_2\; x^2\; +\; \backslash dots\; +\; a\_d\; x^d,$$

or, using the shorter sigma notation:

$$\backslash sum\_\{i=0\}^d\; a\_ix^i.$$

The monomial basis is naturally totally ordered, either by increasing degrees

or by decreasing degrees

$$1\; >\; x\; >\; x^2\; >\; \backslash cdots.$$

In the case of several indeterminates $x\_1,\; \backslash ldots,\; x\_n,$ a monomial is a product

$$x\_1^\{d\_1\}x\_2^\{d\_2\}\backslash cdots\; x\_n^\{d\_n\},$$

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 $1\; =\; x\_1^0\; x\_2^0\backslash cdots\; x\_n^0$ is a monomial.

Similar to the case of univariate polynomials, the polynomials in $x\_1,\; \backslash ldots,\; x\_n$ 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 $d\; =\; d\_1+\backslash cdots+d\_n$ as a basis. The dimension of this subspace is the number of monomials of degree $d$, which is

$$\backslash binom\{d+n-1\}\{d\}\; =\; \backslash frac\{n(n+1)\backslash cdots\; (n+d-1)\}\{d!\},$$

where $\backslash binom\{d+n-1\}\{d\}$ 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

$$\backslash binom\{d\; +\; n\}\{d\}=\; \backslash binom\{d\; +\; n\}\{n\}=\backslash frac\{(d+1)\backslash cdots(d+n)\}\{n!\}.$$

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

and $$1\; \backslash leq\; m$$ for every monomial $m,\; n,\; q.$

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

Category:Algebra

Category:Polynomials