On quadratic Waring’s problem in totally real number fields

Jakub Krásenský, Pavlo Yatsyna

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)
6 Downloads (Pure)


We improve the bound of the g-invariant of the ring of integers of a totally real number field, where the g-invariant g(r) is the smallest number of squares of linear forms in r variables that is required to represent all the quadratic forms of rank r that are representable by the sum of squares. Specifically, we prove that the gOK(r) of the ring of integers OK of a totally real number field K is at most gZ([K : Q]r). Moreover, it can also be bounded by gOF ([K : F]r + 1) for any subfield F of K. This yields a subexponential upper bound for g(r) of each ring of integers (even if the class number is not 1). Further, we obtain a more general inequality for the lattice version G(r) of the invariant and apply it to determine the value of G(2) for all but one real quadratic field.

Original languageEnglish
Pages (from-to)1471-1485
Number of pages15
JournalProceedings of the American Mathematical Society
Issue number4
Publication statusPublished - 1 Apr 2023
MoE publication typeA1 Journal article-refereed


Dive into the research topics of 'On quadratic Waring’s problem in totally real number fields'. Together they form a unique fingerprint.

Cite this