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Abstract
We improve the bound of the ginvariant of the ring of integers of a totally real number field, where the ginvariant 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 g_{OK}(r) of the ring of integers O_{K} of a totally real number field K is at most g_{Z}([K : Q]r). Moreover, it can also be bounded by g_{OF} ([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 

Pages (fromto)  14711485 
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 articlerefereed 
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Dive into the research topics of 'On quadratic Waring’s problem in totally real number fields'. Together they form a unique fingerprint.Projects
 2 Active

Hollanti NT: Numbertheoretic wellrounded lattices
Hollanti, C., Miller, N. & Bolanos Chavez, W.
01/09/2022 → 31/08/2026
Project: Academy of Finland: Other research funding

Hollanti_ICT: Secure Distributed Computation Schemes with Applications to Digitalized Remote Healthcare
Hollanti, C., Villamizar Rubiano, D., Hietaaho, E., Sacikara Kariksiz, E., Yatsyna, P., Kas Hanna, S., Makkonen, O., Matalaaho, T., Karpuk, D. & Allaix, M.
01/01/2021 → 31/12/2023
Project: Academy of Finland: Other research funding