FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 1, PAGES 161-178

A normal form and schemes of quadratic forms

V. M. Levchuk
O. A. Starikova

Abstract

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We present a solution of the problem of the construction of a normal diagonal form for quadratic forms over a local principal ideal ring $R=2R$ with a QF-scheme of order $2$. We give a combinatorial representation for the number of classes of projective congruence quadrics of the projective space over $R$ with nilpotent maximal ideal. For the projective planes, the enumeration of quadrics up to projective equivalence is given; we also consider the projective planes over rings with nonprincipal maximal ideal. We consider the normal form of quadratic forms over the field of $p$-adic numbers. The corresponding QF-schemes have order $4$ or $8$. Some open problems for QF-schemes are mentioned. The distinguished finite QF-schemes of local and elementary types (of arbitrarily large order) are realized as the QF-schemes of a field.

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Last modified: December 21, 2006