Ring of polynomial functions

In mathematics, the ring of polynomial functions on a vector space V over a field k helps us describe polynomial expressions without using specific coordinates. It is written as k[V].

To define this ring, we start with a polynomial ring made from variables t₁, ..., tₙ. These variables act like functions that choose coordinates from points in kₙ. For any vector space V, the ring k[V] becomes a commutative k-algebra from the dual space V∗. This means it is a special kind of subring that includes all polynomials made from the variables linked to the dual space.

When the field k is large (infinite), this ring of polynomial functions matches the symmetric algebra of the dual space V∗_. This idea works even when V comes from a smaller field inside k, like when k is the complex numbers and V is a real vector space. For simplicity, the base field k is assumed to be infinite in this article.

Relation with polynomial ring

Let's see how polynomials and polynomial functions are connected. Imagine two groups: one has all polynomials over a field K, and the other has all polynomial functions in one variable over K. Both groups follow the same rules of multiplication and addition as polynomials.

We can match each polynomial to a function using a special rule. This matching is like a bridge between the two groups. Interestingly, this bridge works perfectly—like a two-way street—only if the field K has infinitely many elements. If K is small, the bridge breaks because some polynomials don’t match up correctly with any function. But when K is large, every polynomial finds its matching function, and the two groups are essentially the same!

Symmetric multilinear maps

Let k be a field and V a finite-dimensional vector space. We look at special functions called symmetric multilinear maps. These maps help us understand polynomial functions on V.

Every symmetric multilinear map can create a homogeneous polynomial function. This means that complicated functions can be built from simpler ones, showing a connection between different areas of mathematics. For example, a bilinear map can create a quadratic form in a unique way.

Taylor series expansion

Main article: Taylor series

When we study smooth functions, we can use a Taylor series to understand the function better. This idea also works for polynomial functions in a vector space. For a polynomial function f, we can see how it changes when we add two points x and y in the space. This gives us a special way to break down the function into simpler parts.

There is a special operator called the polarization operator that helps us study these changes. It shows us how the function grows step by step, which is useful in many areas of mathematics.

Operator product algebra

When we use polynomials with more complex math systems, we can add extra rules to how they work. We might look at functions related to changes instead of just simple numbers.

An operator product algebra is a special math structure. It has rules for how different functions multiply together. These rules help us understand how functions behave when we combine them. They are important in areas like theoretical physics.