M/E29

The famous Langrange's four-square theorem claims that every positive rational integer can be represented as a sum of four squares; Ramanujan later provided a list of all positive definite forms in four variables over Z sharing the same property. In general, for a totally real number field K, a quadratic form over the ring of integers O_K is called universal if it allows to represent every totally positive element of O_K; a notorious example is the sum of three squares for K=Q(\sqrt{5}). Kitaoka conjectured that there are only finitely many fields which admit a ternary universal totally positive definite form; Chan, Kim and Raghavan proved that it is true for classical forms over real quadratic fields. In a recently submitted paper (https://arxiv.org/abs/1909.05422) with M. Tinkova and K. Zemkova, we prove the same for totally real biquadratic fields. In this talk I am going to present the main ideas of the proof and the connection with the so-called indecomposable integers.

Jakub Krasensky

Charles University, Prag