A normal form and schemes of quadratic forms |
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Authors: | V M Levchuk O A Starikova |
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Institution: | (1) Krasnoyarsk State University, Krasnoyarsk, Russia;(2) Northern International University, Magadan, Russia |
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Abstract: | We present a solution of the problem of 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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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 161–178, 2007. |
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