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Deformations of Diophantine systems for quadratic forms of the root lattices \(A_n\). (Russian, English) Zbl 1076.11026
Zap. Nauchn. Semin. POMI 314, 5-14, 285 (2004); translation in J. Math. Sci., New York 133, No. 6, 1605-1610 (2006).
The matrix equation \(Q[X] = {^tT}QT = A\) is studied, where \(Q\) and \(A\) are positive definite symmetric integral matrices of order \(n\) and \(m\), respectively, and \(T\) ranges over the set of \(n\times m\) integer matrices. When \(A = A' \oplus A''\), where the summands have squarefree and coprime determinants, a formula is found for the weighted average of representations of \(A''\) by a suitable quadratic form \(Q'\). This computation depends on the approach due to V. G. Zhuravlev [Algebra Anal. 8 , No. 1, 21–112 (1996; Zbl 0860.11018)]. Two concrete numerical cases are considered in detail.
MSC:
11E12 Quadratic forms over global rings and fields
11D85 Representation problems
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
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