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A second genus of regular ternary forms

Part of: Forms and linear algebraic groups

Published online by Cambridge University Press:  26 February 2010

Irving Kaplansky
Irving Kaplansky
Affiliation:
Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720-5070, U.S.A.
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Extract

In the paper [2] Hsia noted that the forms x2+xy+y2+9z2 and x2+3y2+3yz+3z2 constitute a genus and that both forms are regular; he asked whether there exist any other genera containing two or more regular forms. In this note it is proved that the forms

are regular. They constitute a genus with discriminant 27 (in the normalization used by Brandt and Intrau in [1]). It is noteworthy that Hsia's genus has the same discriminant.

MSC classification

Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
Type
Research Article
Information
Mathematika , Volume 42 , Issue 2 , December 1995 , pp. 444 - 447
Copyright
Copyright © University College London 1995

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References

1.Brandt, H. and Intrau, O.. Tabelle reduzierten positiver tern rer quadratischer Formen. Abh. S chs. Akad. Wiss. Math. Nat. Kl., 45 (1958), no. 4. MR 21, 11493.Google Scholar
2.Hsia, J. S.. Regular positive ternary quadratic forms. Mathematika, 28 (1981), 231–238.CrossRefGoogle Scholar