Almost regular quaternary quadratic forms
Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1499-1549.

We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is p-anisotropic for at most one prime number p. Moreover, for a prime p there is an almost regular p-anisotropic quaternary quadratic form if and only if p37. We also study the genera containing some almost regular p-anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity.

Nous étudions les formes quadratiques entières quaternaires (c’est-à-dire à quatre variables) qui sont définies positives et presque régulières. Nous montrons en particulier qu’une telle forme n’est p-anisotrope que pour au plus un nombre premier p. De plus, pour un nombre premier p, il existe une forme quadratique quaternaire presque régulière p-anisotrope si et seulement si p37. Nous étudions également les genres contenant une forme quadratique presque régulière p-anisotrope. Nous démontrons plusieurs résultats de finitude concernant les familles de ces genres et établissons des critères effectifs presque réguliers.

DOI: 10.5802/aif.2391
Classification: 11E12, 11E20
Keywords: Quadratic equations, almost regular quadratic forms
Mot clés : équations quadratiques, formes quadratiques presque régulières
Bochnak, Jacek 1; Oh, Byeong-Kweon 2

1 Vrije Universiteit Department of Mathematics 1081 HV Amsterdam De Boelelaan 1081 A (The Netherlands)
2 Sejong University Department of Applied Mathematics Seoul, 143-747 (Korea)
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     title = {Almost regular quaternary quadratic forms},
     journal = {Annales de l'Institut Fourier},
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Bochnak, Jacek; Oh, Byeong-Kweon. Almost regular quaternary quadratic forms. Annales de l'Institut Fourier, Volume 58 (2008) no. 5, pp. 1499-1549. doi : 10.5802/aif.2391.

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