Generating sequences and key polynomials

TL;DR

This paper explores various definitions of generating sequences, demonstrating their equivalence under certain conditions, providing counterexamples, and examining their relationship with key polynomials.

Contribution

It clarifies the conditions under which different definitions of generating sequences are equivalent and explores their connection to key polynomials.

Findings

Definitions are equivalent under specific conditions

Counterexample shows they are not always equivalent

Relationship between generating sequences and key polynomials is analyzed

Abstract

The main goal of this paper is to study the different definitions of generating sequences appearing in the literature. We present these definitions and show that under certain situations they are equivalent. We also present an example that shows that they are not, in general, equivalent. We also present the relation of generating sequences and key polynomials.