An upper bound on the number of classes of perfect unary forms in totally real number fields.

Let K be a totally real number field of degree n over ℚ , with discriminant and regulator Δ K , R K , respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by | Δ K | − 1 ∕ 2 γ K n exp (2 n ρ ∞ (Λ K)) , where Δ K is the discriminant of the field K, γ K is the additive Hermite–Humbert constant over positive-definite unary forms for K and ρ ∞ (Λ K) is the covering radius of the log-unit lattice. In particular, when K is Galois over ℚ and n is a prime number, the number of homothety classes of unary forms is upper bounded by (2 3 n) n exp (O (n 3 2 log (n) R K 1 n − 1 )) | Δ K | 1 ∕ 2 , where R K is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by (2 3 n) n exp (O (n 3 2 log (n))) | Δ K | 1 ∕ 2 . [ABSTRACT FROM AUTHOR]