SIC dimension towers via cyclotomic polynomials
Abstract
We prove a structure theorem for the dimension towers~{dk(D)}k≥0 which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~K=Q(qD). If~eps denotes the first totally positive power of a fundamental unit of~K and tk = epsk + eps−k the trace of its k-th power, then the SIC dimension tower~{dk=1+tk}k≥0 is the level~m=3 row of an infinite two-dimensional cyclotomic array~{Ψm(tk)}m≥1,k≥0 attached to~K, while the auxiliary factors~(dk+1) and (dk−3) are its ramified levels m=2 and m=1. Here~Ψm denotes the minimal polynomial of~ ζm+ζm−1 = 2cos2π/m. This construction arose initially from an attempt to formulate relations among SIC dimensions in q-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family cm,k=1−ζmepsk over the cyclotomic field tower {K(μm)}m≥1. This framework sheds new light on the mod-p analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~2mD, the valuations vp(Ψm(tk)) at every fixed level~m are described exactly in terms of a single local unit valuation, which is then related, through the~p-adic class number formula, to the corresponding p-adic L-value.
Related
- Post-critically finite polynomials over cyclotomic closures of number fields
- Cyclotomic Euler–Mahonian polynomials
- Cyclotomic polynomials with trinomial irreducible factors over finite fields
- Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials
- Invariant factors from products of cyclotomic polynomials, with applications
BibTeX
@misc{mcconnell2026dimension,
title = {{SIC dimension towers via cyclotomic polynomials}},
author = {Gary McConnell},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04431},
eprint = {2609.04431},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04431},
}
RIS
TY - UNPB
TI - SIC dimension towers via cyclotomic polynomials
AU - Gary McConnell
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04431
UR - https://arxiv.org/abs/2609.04431
AB - We prove a structure theorem for the dimension towers~{d_k(D)}_k≥0 which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~K=Q(qD). If~eps denotes the first totally positive power of a fundamental unit of~K and t_k = eps^k + eps^−k the trace of its k-th power, then the SIC dimension tower~{d_k=1+t_k}_k≥0 is the level~m=3 row of an infinite two-dimensional cyclotomic array~{Ψ_m(t_k)}_m≥1,k≥0 attached to~K, while the auxiliary factors~(d_k+1) and (d_k−3) are its ramified levels m=2 and m=1. Here~Ψ_m denotes the minimal polynomial of~ ζ_m+ζ_m^−1 = 2cos2π/m. This construction arose initially from an attempt to formulate relations among SIC dimensions in q-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family c_m,k=1−ζ_meps^k over the cyclotomic field tower {K(μ_m)}_m≥1. This framework sheds new light on the mod-p analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~2mD, the valuations v_p(Ψ_m(t_k)) at every fixed level~m are described exactly in terms of a single local unit valuation, which is then related, through the~p-adic class number formula, to the corresponding p-adic L-value.
ER -