Minimal universality criterion sets on the representations of binary quadratic forms

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 S-0 of S is called an S-universality criterion set if any S-0-universal quadratic form is S-universal. We say S-0 is minimal if there does not exist a proper subset of S(0 )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. (C) 2021 Elsevier Inc. All rights reserved.