M/E28

The celebrated 15 theorem of Conway and Schneeberger says that a classical positive definite quadratic form over Z is universal if it represents each element of the set {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim, B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost – and the question of uniqueness for some particular situation has been studied by several authors since. We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. There, the existence of criterion sets for universality has been known, and Lee determined the set for Q(√5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.

Jakub Krásensky

Prag