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
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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 with a QF-scheme
of order .
We give a combinatorial representation for the number of classes
of projective congruence quadrics of the projective space
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
The corresponding QF-schemes have order or .
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
Last modified: December 21, 2006