- Source: 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
1
,
x
,
x
2
,
x
3
,
…
{\displaystyle 1,x,x^{2},x^{3},\ldots }
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
{
1
,
x
,
x
2
,
…
,
x
d
−
1
,
x
d
}
{\displaystyle \{1,x,x^{2},\ldots ,x^{d-1},x^{d}\}}
as a basis.
The canonical form of a polynomial is its expression on this basis:
a
0
+
a
1
x
+
a
2
x
2
+
⋯
+
a
d
x
d
,
{\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{d}x^{d},}
or, using the shorter sigma notation:
∑
i
=
0
d
a
i
x
i
.
{\displaystyle \sum _{i=0}^{d}a_{i}x^{i}.}
The monomial basis is naturally totally ordered, either by increasing degrees
1
<
x
<
x
2
<
⋯
,
{\displaystyle 1
or by decreasing degrees
1
>
x
>
x
2
>
⋯
.
{\displaystyle 1>x>x^{2}>\cdots .}
Several indeterminates
In the case of several indeterminates
x
1
,
…
,
x
n
,
{\displaystyle x_{1},\ldots ,x_{n},}
a monomial is a product
x
1
d
1
x
2
d
2
⋯
x
n
d
n
,
{\displaystyle x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},}
where the
d
i
{\displaystyle d_{i}}
are non-negative integers. As
x
i
0
=
1
,
{\displaystyle 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
⋯
x
n
0
{\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
x
1
,
…
,
x
n
{\displaystyle x_{1},\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
{\displaystyle d}
form a subspace which has the monomials of degree
d
=
d
1
+
⋯
+
d
n
{\displaystyle d=d_{1}+\cdots +d_{n}}
as a basis. The dimension of this subspace is the number of monomials of degree
d
{\displaystyle d}
, which is
(
d
+
n
−
1
d
)
=
n
(
n
+
1
)
⋯
(
n
+
d
−
1
)
d
!
,
{\displaystyle {\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},}
where
(
d
+
n
−
1
d
)
{\textstyle {\binom {d+n-1}{d}}}
is a binomial coefficient.
The polynomials of degree at most
d
{\displaystyle d}
form also a subspace, which has the monomials of degree at most
d
{\displaystyle d}
as a basis. The number of these monomials is the dimension of this subspace, equal to
(
d
+
n
d
)
=
(
d
+
n
n
)
=
(
d
+
1
)
⋯
(
d
+
n
)
n
!
.
{\displaystyle {\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\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
m
<
n
⟺
m
q
<
n
q
{\displaystyle m
and
1
≤
m
{\displaystyle 1\leq m}
for every monomial
m
,
n
,
q
.
{\displaystyle m,n,q.}
See also
Horner's method
Polynomial sequence
Newton polynomial
Lagrange polynomial
Legendre polynomial
Bernstein form
Chebyshev form
Kata Kunci Pencarian:
- Basis (aljabar linear)
- Polinomial monik
- Rentang linear
- Polinomial Newton
- Aljabar nonasosiatif
- Aritmetika modular
- Daftar matriks yang dinamakan
- Monomial basis
- Monomial
- Basis function
- Gröbner basis
- Basis (linear algebra)
- Monomial order
- Standard monomial theory
- Standard basis
- Trigonometric polynomial
- Lagrange polynomial
