## 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
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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).## 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