Work

Irreducibility and locus of complex roots of polynomials related to Fermat’s Last Theorem

Year: 2026

Type: article

Abstract: In this paper, we study the polynomials [Formula: see text] [Formula: see text], whose rational roots would yield counterexamples to Fermat’s Last Theorem. We investigate their factorization over [Formula: see text]. In the case [Formula: see text], we ask whether they are irreducible over [Formula: see text], prove the irreducibility for several infinite families, and investigate the location of the roots of these polynomials on the complex plane. For [Formula: see text], the factorization of [Formula: see text] is intimately related to that of the Cauchy–Mirimanoff polynomials [Formula: see text] and the polynomials [Formula: see text] and [Formula: see text] introduced by Nanninga. After removing the trivial factors [Formula: see text], [Formula: see text], and [Formula: see text], the remaining components agree (up to change of variable) with [Formula: see text], [Formula: see text], or [Formula: see text]. We prove several new irreducibility results for these factors.

Source: International Journal of Number Theory

DOI: 10.1142/s1793042126501174

Authors: Hayk Karapetyan, Ruben Hambardzumyan

Institutions: Yerevan State University, National Academy of Sciences of Armenia

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: State Committee of Science

Awards: G3981498792, G8670978300