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
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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 -