On quadratic Waring’s problem in totally real number fields

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Proceedings of the American Mathematical Society, Volume 151, issue 4
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.
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.
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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