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 Q, with discriminant and regulator ∆ 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 byMoreover, if K is a unit reducible field, the number of classes of perfect unary forms is bound above by O(∆ K exp(2n log(n))).