SIC dimension towers via cyclotomic polynomials

• URL: http://arxiv.org/abs/2609.04431v1
• Date: Thu, 03 Sep 2026 19:51:19 GMT
• Title: SIC dimension towers via cyclotomic polynomials
• Abstract summary: We prove a structure for the dimension towers$d_k(D)_kgeq0$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We prove a structure theorem for the dimension towers~$\{d_k(D)\}_{k\geq0}$ 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~$\{d_k=1+t_k\}_{k\geq0}$ is the level~$m=3$ row of an infinite two-dimensional cyclotomic array~$\{Ψ_m(t_k)\}_{m\geq1,k\geq0}$ 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~ $ζ_m+ζ_m^{-1} = 2\cos{2π/m}$. 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 $c_{m,k}=1-ζ_m\eps^k$ over the cyclotomic field tower $\{K(μ_m)\}_{m\geq1}$. 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.

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