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.

Location: http://mech.math.msu.su/~fpm/eng/k07/k071/k07109h.htm

Last modified: December 21, 2006