TY - JOUR

T1 - ON SMALL VALUES OF INDEFINITE DIAGONAL QUADRATIC FORMS AT INTEGER POINTS IN AT LEAST FIVE VARIABLES

AU - Buterus, Paul

AU - Götze, Friedrich

AU - Hille, Thomas

N1 - Funding Information:
Received by the editors November 22, 2019, and, in revised form, August 26, 2021. 2020 Mathematics Subject Classification. Primary 11D75; Secondary 11J25. Key words and phrases. Irrational quadratic forms, quantitative Oppenheim conjecture. Research funded by DFG (German Research Foundation) - CRC 1283/2 2021 - 317210226.
Publisher Copyright:
© 2022 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0).

PY - 2022/2/11

Y1 - 2022/2/11

N2 - For any ε>0 we derive effective estimates for the size of a nonzero integral point m ∈ Zd \{0} solving the Diophantine inequality |Q[m]| <ε, where Q[m] =q1m21+…+qdm2d denotes a non-singular indefinite diagonal quadratic form in d ≥ 5 variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport to higher dimensions combined with a theorem of Schlickewei. The result obtained is an optimal extension of Schlickewei’s result, giving bounds on small zeros of integral quadratic forms depending on the signature (r, s), to diagonal forms up to a negligible growth factor.

AB - For any ε>0 we derive effective estimates for the size of a nonzero integral point m ∈ Zd \{0} solving the Diophantine inequality |Q[m]| <ε, where Q[m] =q1m21+…+qdm2d denotes a non-singular indefinite diagonal quadratic form in d ≥ 5 variables. In order to prove our quantitative variant of the Oppenheim conjecture, we extend an approach developed by Birch and Davenport to higher dimensions combined with a theorem of Schlickewei. The result obtained is an optimal extension of Schlickewei’s result, giving bounds on small zeros of integral quadratic forms depending on the signature (r, s), to diagonal forms up to a negligible growth factor.

KW - Irrational quadratic forms

KW - quantitative Oppenheim conjecture

UR - http://www.scopus.com/inward/record.url?scp=85123928122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85123928122&partnerID=8YFLogxK

U2 - 10.1090/btran/97

DO - 10.1090/btran/97

M3 - Article

AN - SCOPUS:85123928122

SN - 2330-0000

VL - 9

SP - 1

EP - 34

JO - Transactions of the American Mathematical Society Series B

JF - Transactions of the American Mathematical Society Series B

ER -