Invariant factors from products of cyclotomic polynomials, with applications
Abstract
Let f,g∈ℤ[t] be products of cyclotomic polynomials such that either g/gcd(f,g) or f/gcd(f,g) (or both) is squarefree. We give explicit formulae for the invariant factors of f(Cg), where Cg is the companion matrix of g, expressing them in terms of the indices and multiplicities of the cyclotomic factors of f and g. A more compact formula holds for the circulant case g(t)=tn−1, assuming that every cyclotomic factor shared by f and g has multiplicity $1$ in f. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, K-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when g is squarefree, we determine the least eventual period of the Smith forms of powers of f(Cg).
Related
- Cyclotomic polynomials with trinomial irreducible factors over finite fields
- Post-critically finite polynomials over cyclotomic closures of number fields
- Cyclotomic Euler–Mahonian polynomials
- Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
- SIC dimension towers via cyclotomic polynomials
BibTeX
@misc{noferini2026invariant,
title = {{Invariant factors from products of cyclotomic polynomials, with applications}},
author = {Vanni Noferini and Gerald Williams},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19205},
eprint = {2610.19205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19205},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Invariant factors from products of cyclotomic polynomials, with applications
AU - Vanni Noferini
AU - Gerald Williams
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19205
UR - https://arxiv.org/abs/2610.19205
AB - Let f,g∈ℤ[t] be products of cyclotomic polynomials such that either g/gcd(f,g) or f/gcd(f,g) (or both) is squarefree. We give explicit formulae for the invariant factors of f(C_g), where C_g is the companion matrix of g, expressing them in terms of the indices and multiplicities of the cyclotomic factors of f and g. A more compact formula holds for the circulant case g(t)=t^n−1, assuming that every cyclotomic factor shared by f and g has multiplicity $1$ in f. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, K-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when g is squarefree, we determine the least eventual period of the Smith forms of powers of f(C_g).
ER -