The Hurwitz existence problem and the prime-degree conjecture: A computational perspective

TL;DR

This paper uses computational methods to classify non-realizable partition triples in the Hurwitz existence problem up to degree 31, verifies the prime-degree conjecture for primes less than 32, and introduces a new memory-efficient software architecture.

Contribution

It provides a complete enumeration of non-realizable triples up to degree 31, categorizes them, and proposes a novel software design to handle larger degrees.

Findings

Nearly 90% of non-realizable cases explained by known theory

Verified the prime-degree conjecture for all primes less than 32

Developed a memory-stabilizing computational architecture

Abstract

We investigate the Hurwitz existence problem from a computational viewpoint. Leveraging the symmetric-group algorithm by Zheng and building upon implementations originally developed by Baroni, we achieve a complete and non-redundant enumeration of all non-realizable partition triples for positive integers up to $31$. These results are further categorized into four types according to their underlying mathematical structure; it is observed that nearly nine-tenths of them can be explained by known theoretical results. As an application, we verify the prime-degree conjecture for all primes less than $32$. In light of the exponential memory growth inherent in existing computational approaches -- which limits their feasibility at higher degrees -- we propose a novel software architecture designed to stabilize memory usage, thereby facilitating further detection of exceptional cases in the Hurwitz existence problem. The complete dataset of non-realizable partition triples, along with our implementation, will been made public on GitHub.