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

Abstract

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].