## On quadratic Waring’s problem in totally real number fields

Loading...

##### Journal Title

##### Journal ISSN

##### Volume Title

A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

This publication is imported from Aalto University research portal.

View publication in the Research portal

View/Open full text file from the Research portal

Other link related to publication

View publication in the Research portal

View/Open full text file from the Research portal

Other link related to publication

##### Author

##### Date

2023-04-01

##### Major/Subject

##### Mcode

##### Degree programme

##### Language

en

##### Pages

15

1471-1485

1471-1485

##### Series

Proceedings of the American Mathematical Society, Volume 151, issue 4

##### Abstract

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.##### Description

Funding Information: Received by the editors February 1, 2022, and, in revised form, July 4, 2022, and August 14, 2022. 2020 Mathematics Subject Classification. Primary 11E12, 11D85, 11E25, 11E39. The first author was partially supported by project PRIMUS/20/SCI/002 from Charles University, by Czech Science Foundation GACˇR, grant 21-00420M, by projects UNCE/SCI/022 and GA UK No. 742120 from Charles University, and by SVV-2020-260589. The second author was supported by the project PRIMUS/20/SCI/002 from Charles University and by the Academy of Finland (grants #336005 and #351271, Principal Investigator C. Hollanti). Publisher Copyright: © 2023 American Mathematical Society.

##### Keywords

##### Other note

##### Citation

Krásenský , J & Yatsyna , P 2023 , ' On quadratic Waring’s problem in totally real number fields ' , Proceedings of the American Mathematical Society , vol. 151 , no. 4 , pp. 1471-1485 . https://doi.org/10.1090/proc/16233