Sturm sequences of polynomials related to regular primes

We discuss Kummer’s algebraic definition of a regular prime number p involving the class number of the cyclotomic field Q(ζp) and Kummer’s criterion for a prime number to be regular, which involves the numerators of Bernoulli numbers. A discussion of Sturm sequences and a software implementation follows. We conjecture and provide strong evidence for an integer upper and lower bound, [k − 4 k 5, k − 4 k 6] for the number of real roots of the kth Bernoulli polynomial using data generated by computing Sturm sequences. We then prove explicit formulae for each of the terms of the Sturm sequences of cyclotomic polynomials Φn(x), where n is a prime greater than 3 or a power of 2.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Robinson, Malachi M.

Contributors dc:contributor

• Ahlgren, Scott

• Sturm Sequence

Statement dc:rights

• Copyright 2024 Malachi Robinson

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/127304

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F127304