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Growth of polynomial identities: is the sequence of codimensions eventually non‐decreasing?
Author(s) -
Giambruno Antonio,
Zaicev Mikhail
Publication year - 2014
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdu031
Subject(s) - mathematics , integer (computer science) , sequence (biology) , associative property , polynomial , exponent , zero (linguistics) , associative algebra , field (mathematics) , integer sequence , combinatorics , algebra over a field , discrete mathematics , pure mathematics , division algebra , filtered algebra , mathematical analysis , generating function , genetics , biology , linguistics , philosophy , computer science , programming language
Let A be a PI‐algebra over a field of characteristic zero. Here, we prove that if A is an associative algebra, then the sequence of codimensionsc n ( A ) , n = 1 , 2 , … of A is eventually non‐decreasing. As a consequence, we get thatlim n → ∞log n ( c n ( A ) / exp ( A ) n )exists and is an integer or a half‐integer, where exp ( A ) is the PI‐exponent of A . For the non‐associative case, we construct a non‐associative PI‐algebra B whose sequence of codimensions is not eventually non‐decreasing.