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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Finiteness theorems for positive definite $n$-regular quadratic forms
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by Wai Kiu Chan and Byeong-Kweon Oh
Trans. Amer. Math. Soc. 355 (2003), 2385-2396
DOI: https://doi.org/10.1090/S0002-9947-03-03262-8
Published electronically: January 27, 2003

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).
References
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Bibliographic Information
  • Wai Kiu Chan
  • Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
  • MR Author ID: 336822
  • Email: wkchan@wesleyan.edu
  • 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
  • Email: bkoh@newton.kias.re.kr
  • 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)
  • © Copyright 2003 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 355 (2003), 2385-2396
  • MSC (2000): Primary 11E12, 11E20
  • DOI: https://doi.org/10.1090/S0002-9947-03-03262-8
  • MathSciNet review: 1973994