Monomial basis
In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
One indeterminate
The polynomial ring K[x] of univariate polynomials over a field K is a K-vector space, which has [math]\displaystyle{ 1, x, x^2, x^3, \ldots }[/math] as an (infinite) basis. More generally, if K is a ring then K[x] is a free module which has the same basis.
The polynomials of degree at most d form also a vector space (or a free module in the case of a ring of coefficients), which has [math]\displaystyle{ 1, x, x^2, \ldots }[/math] as a basis.
The canonical form of a polynomial is its expression on this basis: [math]\displaystyle{ a_0 + a_1 x + a_2 x^2 + \dots + a_d x^d, }[/math] or, using the shorter sigma notation: [math]\displaystyle{ \sum_{i=0}^d a_ix^i. }[/math]
The monomial basis is naturally totally ordered, either by increasing degrees [math]\displaystyle{ 1 \lt x \lt x^2 \lt \cdots, }[/math] or by decreasing degrees [math]\displaystyle{ 1 \gt x \gt x^2 \gt \cdots. }[/math]
Several indeterminates
In the case of several indeterminates [math]\displaystyle{ x_1, \ldots, x_n, }[/math] a monomial is a product [math]\displaystyle{ x_1^{d_1}x_2^{d_2}\cdots x_n^{d_n}, }[/math] where the [math]\displaystyle{ d_i }[/math] are non-negative integers. As [math]\displaystyle{ x_i^0 = 1, }[/math] an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular [math]\displaystyle{ 1 = x_1^0 x_2^0\cdots x_n^0 }[/math] is a monomial.
Similar to the case of univariate polynomials, the polynomials in [math]\displaystyle{ x_1, \ldots, x_n }[/math] 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 [math]\displaystyle{ d }[/math] form a subspace which has the monomials of degree [math]\displaystyle{ d = d_1+\cdots+d_n }[/math] as a basis. The dimension of this subspace is the number of monomials of degree [math]\displaystyle{ d }[/math], which is [math]\displaystyle{ \binom{d+n-1}{d} = \frac{n(n+1)\cdots (n+d-1)}{d!}, }[/math] where [math]\displaystyle{ \binom{d+n-1}{d} }[/math] is a binomial coefficient.
The polynomials of degree at most [math]\displaystyle{ d }[/math] form also a subspace, which has the monomials of degree at most [math]\displaystyle{ d }[/math] as a basis. The number of these monomials is the dimension of this subspace, equal to [math]\displaystyle{ \binom{d + n}{d}= \binom{d + n}{n}=\frac{(d+1)\cdots(d+n)}{n!}. }[/math]
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 [math]\displaystyle{ m\lt n \iff mq \lt nq }[/math] and [math]\displaystyle{ 1 \leq m }[/math] for every monomial [math]\displaystyle{ m, n, q. }[/math]
See also
- Horner's method
- Polynomial sequence
- Newton polynomial
- Lagrange polynomial
- Legendre polynomial
- Bernstein form
- Chebyshev form
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