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Transactions of the American Mathematical Society

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Finiteness theorems for positive definite $n$-regular quadratic forms

Authors: Wai Kiu Chan and Byeong-Kweon Oh
Journal: Trans. Amer. Math. Soc. 355 (2003), 2385-2396
MSC (2000): Primary 11E12, 11E20
Published electronically: January 27, 2003
MathSciNet review: 1973994
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Abstract: An integral quadratic form $f$ of $m$ variables is said to be $n$-regular if $f$ globally represents all quadratic forms of $n$ variables that are represented by the genus of $f$. For any $n \geq 2$, it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of $n + 3$variables that are $n$-regular. We also investigate similar finiteness results for almost $n$-regular and spinor $n$-regular quadratic forms. It is shown that for any $n \geq 2$, there are only finitely many equivalence classes of primitive positive definite spinor or almost $n$-regular quadratic forms of $n + 2$ variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

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Additional Information

Wai Kiu Chan
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459

Byeong-Kweon Oh
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Address at time of publication: School of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheongyangri-dong, Dongdaemun-gu Seoul 130-012, Korea

Keywords: Regular integral quadratic forms
Received by editor(s): July 13, 2002
Received by editor(s) in revised form: November 19, 2002
Published electronically: January 27, 2003
Additional Notes: The research of the first author is partially supported by the National Security Agency and the National Science Foundation
The second author was supported by a postdoctoral fellowship program from the Korea Science and Engineering Foundation (KOSEF)
Article copyright: © Copyright 2003 American Mathematical Society