Minimal universality criterion sets on the representations of binary quadratic forms

Abstract

For a set S of (positive definite and integral) quadratic forms with bounded rank, a quadratic form f is called S-universal if it represents all quadratic forms in S. A subset S0 of S is called an S-universality criterion set if any S0-universal quadratic form is S-universal. We say S0 is minimal if there does not exist a proper subset of S0 that is an S-universality criterion set. In this article, we study various properties of minimal universality criterion sets. In particular, we show that for ‘most’ binary quadratic forms f, minimal S-universality criterion sets are unique in the case when S is the set of all subforms of the binary form f.