# Inverse-producing extensions of Topological Algebras/Algebra of polynomials

## Introduction

A polynomial algebra ${\displaystyle A[t]}$ is a vector space of polynomials, where the coefficients come from the given algebra ${\displaystyle A}$. The polynomial algebra ${\displaystyle A[t]}$ is an essential tool to construct an algebra extension ${\displaystyle B}$ of ${\displaystyle A}$ in which a given ${\displaystyle z\in A}$ is invertible if it satisfies certain topological invertibility criteria.

### Remark: Algebra expansion

In the construction of algebra extension in which a ${\displaystyle z\in A}$ is invertible, the first step is to consider the algebra of polynomials ${\displaystyle A[t]}$. The following figure shows how the algebra extension ${\displaystyle B}$ is constructed over the polynomial algebra.

## Algebraic closure

We only extend the algebra ${\displaystyle A}$ to contain an additional element ${\displaystyle t\notin A}$, which is to be contained in an algebra extension ${\displaystyle B}$ of ${\displaystyle A}$. Since multiplication and addition must be completed in ${\displaystyle B}$, polynomials result from multiplications ${\displaystyle a\cdot t}$ and ${\displaystyle t\cdot t=t^{2}}$ with coefficients ${\displaystyle a\in A}$n, which must be contained as summands ${\displaystyle a_{n}\cdot t^{n}}$ as polynomials in the algebra expansion.

### Extension of the algebra

This implies the closure of the

• multiplicative linkage of ${\displaystyle t}$ with itself and therefore ${\displaystyle t^{n}}$ with ${\displaystyle n\in \mathbb {N} }$ must also be in ${\displaystyle B}$ again,
• the arbitrary multiplicative links of ${\displaystyle t^{n}\in B}$ with elements from ${\displaystyle A}$ lie again in ${\displaystyle B}$, i.e. ${\displaystyle a_{n}\cdot t^{n}\in B}$.
• the additive algebraic algebraic closure also eventually requires that additive links from ${\displaystyle a_{n}\cdot t^{n}\in B}$ lie again in ${\displaystyle B}$.

### Polynomials with coefficients from the given algebra

Out of this necessity, one considers polynomials with coefficients from ${\displaystyle A}$ as the first step in constructing an algebra expansion in which a ${\displaystyle z}$ can be invertible.

## Polynomial algebra

We now consider, for a given topological algebra ${\textstyle \left(A,\left\|\cdot \right\|_{\mathcal {A}}\right)\in {\mathcal {K}}(\mathbb {K} )}$, the set of polynomials with coefficients in ${\displaystyle A}$.

${\displaystyle p(t)=\sum _{k=0}^{n}p_{k}\cdot t^{k}{\mbox{ with }}p_{k}\in A{\mbox{ for }}k\in \{0,1,...,n\}}$

and power series with coefficients in algebra ${\displaystyle A}$.

${\displaystyle p(t)=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ with }}p_{k}\in A{\mbox{ for }}k\in \mathbb {N} _{o}}$

### Degree of polynomials

First of all, polynomials would be more formally notated in the above form with ${\displaystyle n\in \mathbb {N} _{o}}$ and with ${\displaystyle p_{n}\not =0_{A}}$ ${\displaystyle n}$ would indicate the degree of the polynomial. For the Cauchy product of two polynomials ${\displaystyle p,q\in A[t]}$, however, this notation is unsuitable, since in the addition and multiplication two polynomials ${\displaystyle p,q\in A[t]}$ the handling of the degree entails additional formal overhead, which, however, does not matter for the further considerations of algebra expansions.

### Notation for the polynomial algebra

Therefore, polynomials are defined as follows over "finite" sequences ${\displaystyle c_{oo}(A)}$, which from an index bound ${\displaystyle n\in \mathbb {N} _{o}}$ consists only of the zero vector ${\displaystyle 0_{A}}$ in ${\displaystyle A}$.

${\displaystyle p(t)=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ with }}(p_{k})_{k\in \mathbb {N} _{0}}\in c_{oo}(A)}$

## Cauchy product

Given, in general, two polynomials ${\displaystyle p,q}$ with coefficients from ${\textstyle A}$.

${\displaystyle p(t):=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ und }}q(t):=\sum _{k=0}^{\infty }q_{k}\cdot t^{k}}$

Then Cauchy product of ${\displaystyle p,q}$ is defined as follows:

${\displaystyle p(t)\cdot q(t):=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{n}p_{k}\cdot q_{n-k}\right)t^{n}.}$