The exact value of the centralizer dimension is found for a free polynilpotent group and for a free group in a variety of metabelian groups of nilpotency class at most c. Relations between ∃- and Φ-theories of groups are specified, in which case the concept of centralizer dimension plays an important role.

中文翻译：

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群的普遍理论和中心化维度

对于自由多幂零群和至多为 c 的幂零类元贝群中的自由群，可以找到中心化维数的精确值。指定了群的 ∃- 和 Φ- 理论之间的关系，在这种情况下，集中器维数的概念起着重要的作用。