Ring of polynomial functions - Alchetron, the free social encyclopedia

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 k [ t 1 , … , t n ] is a polynomial ring, then we can view t i as coordinate functions on k n ; i.e., t i ( x ) = x i when x = ( x 1 , … , x n ) . This suggests the following: given a vector space V, let k[V] be the subring generated by the dual space V ∗ of the ring of all functions V → k . If we fix a basis for V and write t i for its dual basis, then k[V] consists of polynomials in t i ; 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 k be an infinite field of characteristic zero (or at least very large) and V a finite-dimensional vector space.

Let S q ( V ) denote the vector space of multilinear functionals λ : ∏ 1 q V → k that are symmetric; λ ( v 1 , … , v q ) is the same for all permutations of v i 's.

Any λ in S q ( V ) gives rise to a homogeneous polynomial function f of degree q: we just let f ( v ) = λ ( v , … , v ) . To see that f is a polynomial function, choose a basis e i , 1 ≤ i ≤ n of V and t i its dual. Then

λ ( v 1 , … , v q ) = ∑ i 1 , … , i q = 1 n λ ( e i 1 , … , e i q ) t i 1 ( v 1 ) ⋯ t i q ( v q ) ,

which implies f is a polynomial in t_i's.

Thus, there is a well-defined linear map:

ϕ : S q ( V ) → k [ V ] q , ϕ ( λ ) ( v ) = λ ( v , ⋯ , v ) .

We show it is an isomorphism. Choosing a basis as before, any homogeneous polynomial function f of degree q can be written as:

f = ∑ i 1 , … , i q = 1 n a i 1 ⋯ i q t i 1 ⋯ t i q

where a i 1 ⋯ i q are symmetric in i 1 , … , i q . Let

ψ ( f ) ( v 1 , … , v q ) = ∑ i 1 , ⋯ , i q = 1 n a i 1 ⋯ i q t i 1 ( v 1 ) ⋯ t i q ( v q ) .

Clearly, φ ∘ ψ is the identity; in particular, φ is surjective. To see φ is injective, suppose φ(λ) = 0. Consider

ϕ ( λ ) ( t 1 v 1 + ⋯ + t q v q ) = λ ( t 1 v 1 + ⋯ + t q v q , . . . , t 1 v 1 + ⋯ + t q v q ) ,

which is zero. The coefficient of t₁t₂ … t_q in the above expression is q! times λ(v₁, …, v_q); it follows that λ = 0.

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.

Taylor series expansion

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,

f ( x + y ) = ∑ n = 0 ∞ g n ( x , y )

where g_n(x, y) are homogeneous of degree n in y and only finitely many of them are nonzero. We then let

( P y f ) ( x ) = g 1 ( x , y ) ,

resulting in the linear endomorphism P_y of k[V]. It is called the polarization operator. We then have, as promised:

Proof: We first note that (P_y f) (x) is the coefficient of t in f(x + t y); in other words, since g₀(x, y) = g₀(x, 0) = f(x),

P y f ( x ) = d d t | t = 0 f ( x + t y )

where the right-hand side is, by definition,

f ( x + t y ) − f ( x ) t | t = 0 .

The theorem follows from this. For example, for n = 2, we have:

P y 2 f ( x ) = ∂ ∂ t 1 | t 1 = 0 P y f ( x + t 1 y ) = ∂ ∂ t 1 | t 1 = 0 ∂ ∂ t 2 | t 2 = 0 f ( x + ( t 1 + t 2 ) y ) = 2 ! g 2 ( x , y ) .

The general case is similar. ◻

Operator product algebra

When the polynomials are valued not over a field k, but instead are valued over some algebra, then one may define additional structure. Thus, for example, one may consider the ring of functions over GL(n,m), instead of for k = GL(1,m). In this case, one may impose an additional axiom.

The operator product algebra is an associative algebra of the form

A i ( x ) B j ( y ) = ∑ k f k i j ( x , y , z ) C k ( z )

The structure constants f k i j ( x , y , z ) are required to be single-valued functions, rather than sections of some vector bundle. The fields (or operators) A i ( x ) 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 | x − y | . Thus, the ring of functions can be taken to be the ring of polynomial functions.

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.

References

Ring of polynomial functions Wikipedia

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Contents

Symmetric multilinear maps

Taylor series expansion

Operator product algebra

References