Ring of polynomial functions

In mathematics, the ring of polynomial functions on a vector space V over a field k provides a way to describe polynomial expressions without needing to refer to specific coordinates. It is written as k[V]. When V has a limited number of dimensions and is seen as an algebraic variety, this ring matches exactly with the coordinate ring of V.

To define this ring clearly, we start with a polynomial ring made from variables t₁, ..., tₙ. These variables can be thought of as functions that pick out coordinates from points in kₙ. By using this idea, for any vector space V, the ring k[V] becomes a commutative k-algebra created from the dual space V∗. This means it is a special kind of subring that includes all polynomials formed from the variables linked to the dual space.

When the field k is large (infinite), this ring of polynomial functions is the same as the symmetric algebra of the dual space V∗_. In practical uses, this definition works even when V comes from a smaller field inside k, such as when k is the complex numbers and V is a real vector space. For easier discussion, the base field k is assumed to be infinite throughout the article.

Relation with polynomial ring

Let’s explore how polynomials and polynomial functions are connected. Imagine two sets: one contains all polynomials over a field K, and the other contains all polynomial functions in one variable over K. Both sets 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 sets. 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 sets 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 deep 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 something called a Taylor series to understand the function better. This idea also works for polynomial functions on a vector space. For a polynomial function f, we can look at 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 lets us see how the function grows step by step, which is useful in many areas of mathematics.

Operator product algebra

When polynomials are not used with normal numbers but with more complex mathematical systems, we can add extra rules to how they work. For example, we might look at functions related to certain transformations instead of simple numbers.

An operator product algebra is a special type of mathematical structure where we have rules for how different functions multiply together. These rules help us understand how functions behave when we combine them, and they are important in areas like theoretical physics.