In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring.
If V is finite dimensional 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.
for its dual basis, then k[V] consists of polynomials in
If k is infinite, then k[V] is the symmetric algebra of the dual space
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.)
Throughout the article, for simplicity, the base field k is assumed to be infinite.
A routine check shows that the mapping
This homomorphism is an isomorphism if and only if K is an infinite field.
is the zero function and our homomorphism is not an isomorphism (and, actually, the algebras are not isomorphic, since the algebra of polynomials is infinite while that of polynomial functions is finite).
Since this mapping is clearly surjective, it is bijective and thus an algebra isomorphism of A and B.
Let k be an infinite field of characteristic zero (or at least very large) and V a finite-dimensional vector space.
denote the vector space of multilinear functionals
gives rise to a homogeneous polynomial function f of degree q: we just let
To see that f is a polynomial function, choose a basis
Thus, there is a well-defined linear map: We show it is an isomorphism.
Choosing a basis as before, any homogeneous polynomial function f of degree q can be written as: where
Note: φ is independent of a choice of basis; so the above proof shows that ψ is also independent of a basis, the fact not a priori obvious.
Example: A bilinear functional gives rise to a quadratic form in a unique way and any quadratic form arises in this way.
Given a smooth function, locally, one can get a partial derivative of the function from its Taylor series expansion and, conversely, one can recover the function from the series expansion.
This fact continues to hold for polynomials functions on a vector space.
If f is in k[V], then we write: for x, y in V, where gn(x, y) are homogeneous of degree n in y, and only finitely many of them are nonzero.
We then let resulting in the linear endomorphism Py of k[V].
We then have, as promised: Theorem — For each f in k[V] and x, y in V, Proof: We first note that (Py f) (x) is the coefficient of t in f(x + t y); in other words, since g0(x, y) = g0(x, 0) = f(x), where the right-hand side is, by definition, The theorem follows from this.
When the polynomials are valued not over a field k, but over some algebra, then one may define additional structure.
[clarification needed] In this case, one may impose an additional axiom.
are required to be single-valued functions, rather than sections of some vector bundle.
are required to span the ring of functions.
In practical calculations, it is usually required that the sums be analytic within some radius of convergence; typically with a radius of convergence of
The above can be considered to be an additional requirement imposed on the ring; it is sometimes called the bootstrap.
In physics, a special case of the operator product algebra is known as the operator product expansion.