Geodesic
Enumeration of quadrics of projective spaces over local rings
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2011), pp. 59-63.

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In the paper in Comm. Algebra 36, 1426–1436 (2008), using integral representation methods for combinatorial sums, G. P. Egorychev and E. V. Zima have obtained simple formulas for the number of classes of projectively congruent quadrics and posed a question on an algebraic proof and an interpretation of the obtained formulas. The goal of this paper is to give an answer to this question.
Keywords: projective space, local ring, quadrics, symmetric forms, projective congruence. 
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O. A. Starikova; A. V. Svistunova. Enumeration of quadrics of projective spaces over local rings. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 12 (2011), pp. 59-63. http://geodesic.mathdoc.fr/item/IVM_2011_12_a6/

[1] Ojanguren M., Sridharan R., “A note on the fundamental theorem of projective geometry”, Comm. Math. Helv., 44:3 (1969), 3100–3150 | MR

[2] Benz W., Vorlesungen über Geometrie der Algebren, Springer-Verlag, Berlin etc., 1973 | MR | Zbl

[3] Vishnevskii V. V., Rozenfeld B. A., Shirokov A. P., “O razvitii geometrii prostranstv nad algebrami”, Izv. vuzov. Matem., 1984, no. 7, 38–44 | MR | Zbl

[4] Milnor Dzh., Khyuzmoller D., Simmetricheskie bilineinye formy, Nauka, M., 1986 | MR

[5] Levchuk V. M., Starikova O. A., “Normalnyi vid i skhemy kvadratichnykh form”, Fundament. i prikl. matem., 13:1 (2007), 161–178 | MR | Zbl

[6] Levchuk V. M., Starikova O. A., “Kvadratichnye formy proektivnykh prostranstv nad koltsami”, Matem. sb., 197:6 (2006), 97–110 | MR | Zbl

[7] Egorychev G. P., Zima E. V., “Simple formulae for the number of quadrics and symmetric forms of modules over local rings”, Comm. in Algebra, 36 (2008), 1426–1436 | DOI | MR | Zbl