2022/06/23 by Ján Mináč, Mináč, Ján, Tung Nguyen +3 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2206.11778
openalex publication_date 2022/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For each prime number p one can associate a Fekete polynomial with coefficients -1 or 1 except the constant term, which is 0. These are classical polynomials that have been studied extensively in the framework of analytic number theory. In a recent paper, we showed that these polynomials also encode interesting arithmetic information. In this paper, we define generalized Fekete polynomials associated with quadratic characters whose conductors could be a composite number. We then investigate the appearance of cyclotomic factors of these generalized Fekete polynomials. Based on this investigation, we introduce a compact version of Fekete polynomials as well as their trace polynomials. We then study the Galois groups of these Fekete polynomials using modular techniques. In particular, we discover some surprising extra symmetries which imply some restrictions on the corresponding Galois groups. Finally, based on both theoretical and numerical data, we propose a precise conjecture on the structure of these Galois groups.