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

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

### Remark: Algebra expansion

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

## Algebraic closure

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

### Extension of the algebra

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

### Polynomials with coefficients from the given algebra

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

## Polynomial algebra

We now consider, for a given topological algebra (A, ‖⋅‖$_A$)∈K(K){\textstyle \left(A, \left\|\cdot \right\|_{\mathcal {A}}\right)\in {\mathcal {K}}(\mathbb {K} )}, the set of polynomials with coefficients in $A$.
    $p(t)=\sum _{k=0}^{n}p_{k}\cdot t^{k}{\mbox{ with }}p_{k}\in A{\mbox{ for }}k\in \{0,1,...,n\}$
and power series with coefficients in algebra $A$.
    $p(t)=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ with }}p_{k}\in A{\mbox{ for }}k\in \mathbb {N} _{o}$

### Degree of polynomials

First of all, polynomials would be more formally notated in the above form with $n\in \mathbb {N} _{o}$ and with $p_{n}\not =0_{A}$ $n$ would indicate the degree of the polynomial. For the Cauchy product of two polynomials $p,q\in A[t]$, however, this notation is unsuitable, since in the addition and multiplication two polynomials $p,q\in A[t]$ 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 $c_{oo}(A)$, which from an index bound $n\in \mathbb {N} _{o}$ consists only of the zero vector $0_{A}$ in $A$.
    $p(t)=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ with }}(p_{k})_{k\in \mathbb {N} _{0}}\in c_{oo}(A)$

## Cauchy product

Given, in general, two polynomials $p,q$ with coefficients from A{\textstyle A}.
    $p(t):=\sum _{k=0}^{\infty }p_{k}\cdot t^{k}{\mbox{ und }}q(t):=\sum _{k=0}^{\infty }q_{k}\cdot t^{k}$
Then Cauchy product of $p,q$ is defined as follows:
    $p(t)\cdot q(t):=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{n}p_{k}\cdot q_{n-k}\right)t^{n}.$

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