Ring of polynomial functions
In mathematics, the ring of polynomial functions on a vector space V over an infinite field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V has finite dimension and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V.
The explicit definition of the ring can be given as follows. If is a polynomial ring, then we can view as coordinate functions on ; i.e., when This suggests the following: given a vector space V, let k[V] be the subring generated by the dual space of the ring of all functions . If we fix a basis for V and write for its dual basis, then k[V] consists of polynomials in ; it is a polynomial ring.
In applications, one also defines k[V] when V is defined over some subfield of k (e.g., k is the complex field and V is a real vector space.) The same definition still applies.
Symmetric multilinear maps
Let denote the vector space of multilinear linear functionals that are symmetric; is the same for all permutations of 's.
Any λ in gives rise to a homogeneous polynomial function f of degree q: let To see that f is a polynomial function, choose a basis of V and its dual. Then
Thus, there is a well-defined linear map:
It is an isomorphism: choosing a basis as before, any homogeneous polynomial function f of degree q can be written as:
where are symmetric in . Let
Then ψ is the inverse of φ. (Note: φ is still independent of a choice of basis; so ψ is also independent of a basis.)
Example: A bilinear functional gives rise to a quadratic form in a unique way and any quadratic form arises in this way.
- There is also a more abstract way to see this: to give a multilinear functional on the product of q copies of V is the same as to give a linear functional on the q-th tensor power of V. The requirement that the multilinear functional to be symmetric translates to the one that the linear functional on the tensor power factors through the q-th symmetric power of V, which is isomorphic to k[V]q.
- Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, Vol. 2 (new ed.), Wiley-Interscience (published 2004).
|This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.|