Inverse-producing extensions of Topological Algebras/Algebra of polynomials

Introduction

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

Remark: Algebra expansion

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In the construction of algebra extension in which a   is invertible, the first step is to consider the algebra of polynomials  . The following figure shows how the algebra extension   is constructed over the polynomial algebra.

 

Algebraic closure

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We only extend the algebra   to contain an additional element  , which is to be contained in an algebra extension   of  . Since multiplication and addition must be completed in  , polynomials result from multiplications   and   with coefficients  n, which must be contained as summands   as polynomials in the algebra expansion.

Extension of the algebra

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This implies the closure of the

  • multiplicative linkage of   with itself and therefore   with   must also be in   again,
  • the arbitrary multiplicative links of   with elements from   lie again in  , i.e.  .
  • the additive algebraic algebraic closure also eventually requires that additive links from   lie again in  .

Polynomials with coefficients from the given algebra

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

Polynomial algebra

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We now consider, for a given topological algebra  , the set of polynomials with coefficients in  .

 

and power series with coefficients in algebra  .

 

Degree of polynomials

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First of all, polynomials would be more formally notated in the above form with   and with     would indicate the degree of the polynomial. For the Cauchy product of two polynomials  , however, this notation is unsuitable, since in the addition and multiplication two polynomials   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

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Therefore, polynomials are defined as follows over "finite" sequences  , which from an index bound   consists only of the zero vector   in  .

 

Cauchy product

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Given, in general, two polynomials   with coefficients from  .

 

Then Cauchy product of   is defined as follows:

 

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