Multiplicative summations into algebraically closed fields

TL;DR

This paper develops a framework for extending multiplicative summations into algebraically closed fields, introducing scalar polynomials and minimal polynomials to analyze series and their mappings.

Contribution

It introduces maximal canonical extensions and the concept of scalar polynomials as analogues to minimal polynomials for series over rings.

Findings

Defined scalar polynomials for series algebraic over rings.

Established the uniqueness of the value for series with scalar polynomial of form (t-a)^n.

Extended the theory of summations into algebraically closed fields.

Abstract

In this paper, extending our earlier program, we derive maximal canonical extensions for multiplicative summations into algebraically closed fields. We show that there is a well-defined analogue to minimal polynomials for a series algebraic over a ring of series, the "scalar polynomial". When that ring is the domain of a summation $\mathfrak{S}$, we derive the related concepts of the $\mathfrak{S}$-minimal polynomial for a series, which is mapped by $\mathfrak{S}$ to a scalar polynomial. When the scalar polynomial for a series has the form $(t-a)^n$, $a$ is the unique value to which the series can be mapped by an extension of the original summation.