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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.
Original language | English |
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Pages (from-to) | 1471-1485 |
Number of pages | 15 |
Journal | Proceedings of the American Mathematical Society |
Volume | 151 |
Issue number | 4 |
DOIs | |
Publication status | Published - 1 Apr 2023 |
MoE publication type | A1 Journal article-refereed |
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Hollanti NT: Number-theoretic well-rounded lattices
Hollanti, C. (Principal investigator), Miller, N. (Project Member), Bolanos Chavez, W. (Project Member), Matala-aho, T. (Project Member) & Forst, M. (Project Member)
01/09/2022 → 31/08/2026
Project: Academy of Finland: Other research funding
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Hollanti_ICT: Secure Distributed Computation Schemes with Applications to Digitalized Remote Healthcare
Hollanti, C. (Principal investigator), Villamizar Rubiano, D. (Project Member), Hieta-aho, E. (Project Member), Sacikara Kariksiz, E. (Project Member), Yatsyna, P. (Project Member), Kas Hanna, S. (Project Member), Makkonen, O. (Project Member), Matala-aho, T. (Project Member), Karpuk, D. (Project Member), Bolanos Chavez, W. (Project Member) & Allaix, M. (Project Member)
01/01/2021 → 31/12/2023
Project: Academy of Finland: Other research funding