Abstract
Let K be a number field that is Galois over Q with unit group of rank d. We obtain an upper bound on the number of facets of the reduction domain (and therefore the fundamental domain) of K of O(((d+1)ρ∞(ΛK)+log(d2))d(d+1)!), where ρ∞(ΛK) is the covering radius of the log-unit lattice in the infinity norm. As a side-result, we also show that the Weil height of the shortest Pisot unit in the number field can be no greater than γ[K:Q]((d+1)ρ∞(ΛK)+dϵ) where γ=1 if K is totally real or 2 otherwise and ϵ>0 is some arbitrarily small constant.
Published Version
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