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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
##### Keywords:
embeddings of lattices; weight of representations
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