Enumeration of quadrics of projective spaces over local rings

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.