On quadratic Waring’s problem in totally real number fields

Jakub Krásenský; Pavlo Yatsyna

doi:10.1090/proc/16233

On quadratic Waring’s problem in totally real number fields

Jakub Krásenský, Pavlo Yatsyna

Charles University

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

1 Sitaatiot (Scopus)

48 Lataukset (Pure)

Abstrakti

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 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.

Alkuperäiskieli	Englanti
Sivut	1471-1485
Sivumäärä	15
Julkaisu	Proceedings of the American Mathematical Society
Vuosikerta	151
Numero	4
DOI - pysyväislinkit	https://doi.org/10.1090/proc/16233
Tila	Julkaistu - 1 huhtik. 2023
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Pääsy asiakirjaan

10.1090/proc/16233

SCI_Krasensky_etal_Proceedings_of_the_AMS_2023Accepted author manuscript, 218 KBLisenssi: CC BY

Muut tiedostot ja linkit

Linkki julkaisuun Scopuksessa

1 Päättynyt
1 Aktiivinen

Hollanti NT: Number-theoretic well-rounded lattices
Hollanti, C. (Vastuullinen tutkija), Miller, N. (Projektin jäsen), Bolaños, W. (Projektin jäsen), Matala-aho, T. (Projektin jäsen) & Forst, M. (Projektin jäsen)
01/09/2022 → 31/08/2026
Projekti: Academy of Finland: Other research funding
Hollanti_ICT: Secure Distributed Computation Schemes with Applications to Digitalized Remote Healthcare
Hollanti, C. (Vastuullinen tutkija), Villamizar Rubiano, D. (Projektin jäsen), Hieta-aho, E. (Projektin jäsen), Sacikara, E. (Projektin jäsen), Yatsyna, P. (Projektin jäsen), Kas Hanna, S. (Projektin jäsen), Makkonen, O. (Projektin jäsen), Matala-aho, T. (Projektin jäsen), Karpuk, D. (Projektin jäsen), Bolaños, W. (Projektin jäsen) & Allaix, M. (Projektin jäsen)
01/01/2021 → 31/12/2023
Projekti: Academy of Finland: Other research funding

Siteeraa tätä

@article{59aaeb2cdfdd46bf8c68ac65a790b3d6,

title = "On quadratic Waring{\textquoteright}s problem in totally real number fields",

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.",

author = "Jakub Kr{\'a}sensk{\'y} and Pavlo Yatsyna",

note = "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: {\textcopyright} 2023 American Mathematical Society.",

year = "2023",

month = apr,

day = "1",

doi = "10.1090/proc/16233",

language = "English",

volume = "151",

pages = "1471--1485",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "4",

}

TY - JOUR

T1 - On quadratic Waring’s problem in totally real number fields

AU - Krásenský, Jakub

AU - Yatsyna, Pavlo

N1 - 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.

PY - 2023/4/1

Y1 - 2023/4/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85149247277&partnerID=8YFLogxK

U2 - 10.1090/proc/16233

DO - 10.1090/proc/16233

M3 - Article

AN - SCOPUS:85149247277

SN - 0002-9939

VL - 151

SP - 1471

EP - 1485

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 4

ER -

On quadratic Waring’s problem in totally real number fields

Abstrakti

Pääsy asiakirjaan

Muut tiedostot ja linkit

Sormenjälki

Projektit

Hollanti NT: Number-theoretic well-rounded lattices

Hollanti_ICT: Secure Distributed Computation Schemes with Applications to Digitalized Remote Healthcare

Siteeraa tätä