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 Formula: see text, with discriminant and regulator Formula: see text, 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 Formula: see text, where Formula: see text is the discriminant of the field K, Formula: see text is the additive Hermite–Humbert constant over positive-definite unary forms for K and Formula: see text is the covering radius of the log-unit lattice. In particular, when K is Galois over Formula: see text and n is a prime number, the number of homothety classes of unary forms is upper bounded by Formula: see text, where Formula: see text 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 Formula: see text.