Essentially Unique Representations by Certain Ternary Quadratic Forms

Abstract

In this paper, we generalize the idea of “essentially unique” representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers that are represented in essentially one way by a given form that is alone in its genus. We consider a variety of forms that illustrate how this method applies to each of the 794 ternary quadratic forms that are alone in their genus. As a consequence, we resolve some conjectures of Kaplansky regarding unique representation by the forms x² + y² + 3z², x² + 3y² + 3z², and x² + 2y² + 3z².

2000 AMS Subject Classification:

• 11B65
• 11E16
• 11E20
• 11E25
• 11E41
• 11F37

Keywords:

• representations of integers
• sum of three squares
• local densities
• ternary quadratic forms
• Siegel formula