Elementary characters on semigroups: the rational case

TL;DR

This paper investigates which characters on polynomials can be extended to rational functions, conjecturing that only elementary characters depending on degree can be extended, and provides counterexamples of non-extendable characters.

Contribution

It proposes a conjecture characterizing extendable characters and constructs explicit examples of non-extendable characters on polynomials.

Findings

Conjecture that only elementary characters extend to rational functions

Constructed examples of non-elementary characters not extendable

Clarified the relationship between polynomial and rational function characters

Abstract

Since polynomials form a subsemigroup of the semigroup of rational functions, every character on rational functions is a character on polynomials. On the other direction, not every character on polynomials is the restriction of a character on rational functions. What are the characters on polynomials that can be extended to rational functions? In this work, we conjecture that the only characters that can be extended are those that depends on the degree, often called elementary. Also, we construct two example of character on polynomials, not elementaries, that cannot be extended to rational functions.