Endomorphisms of a certain ring of rational functions

TL;DR

This paper introduces a specialized ring of rational functions with unique operations and classifies its endomorphisms, revealing a structured family and degenerations linked to roots of unity.

Contribution

It defines a novel ring with unconventional operations and provides a comprehensive classification of its endomorphisms, including degenerations related to roots of unity.

Findings

Endomorphisms form a well-structured family or degenerate.

Degenerations are characterized by roots of unity.

Complete classification of endomorphisms achieved.

Abstract

We define a ring whose elements are rational functions, whose addition is polynomial multiplication, and whose multiplication is a convolution operation. It is then show that this ring's endomorphisms exhibit a strong classification. Namely, they will all fall in a well structured family, or they will degenerate. This degeneration is classified strongly in terms of roots of unity.