Growth of polynomial identities: is the sequence of codimensions eventually non‐decreasing? | Zendy

Giambruno Antonio | Zendy; Zaicev Mikhail | Zendy

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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.

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