Let V be a variety of non necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions cn(V); n = 1; 2, … and here we study varieties of polynomial growth. Recently, for any real number a, 3 &lt; a &lt; 4, a variety V was constructed satisfying C1n^a &lt; cn(V) &lt; C2n^a; for some constants C1;C2. Motivated by this result here we try to classify all possible growth of varieties V such that cn(V) &lt; Cn^a; with 0 &lt; a &lt; 2, for some constant C. We prove that if 0 &lt; a &lt; 1 then, for n large, cn(V) ≤ 1, whereas if V is a commutative variety and 1 &lt; a &lt; 2, then lim logn cn(V) = 1 or cn(V) ≤ 1 for n large enough.

MISHCHENKO, S., VALENTI, A. (2010). Varieties with at most quadratic growth. ISRAEL JOURNAL OF MATHEMATICS, 178(1), 209-228 [10.1007/s11856-010-0063-4].

Varieties with at most quadratic growth

Abstract

Let V be a variety of non necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions cn(V); n = 1; 2, … and here we study varieties of polynomial growth. Recently, for any real number a, 3 < a < 4, a variety V was constructed satisfying C1n^a < cn(V) < C2n^a; for some constants C1;C2. Motivated by this result here we try to classify all possible growth of varieties V such that cn(V) < Cn^a; with 0 < a < 2, for some constant C. We prove that if 0 < a < 1 then, for n large, cn(V) ≤ 1, whereas if V is a commutative variety and 1 < a < 2, then lim logn cn(V) = 1 or cn(V) ≤ 1 for n large enough.
Scheda breve Scheda completa Scheda completa (DC)
2010
Settore MAT/02 - Algebra
MISHCHENKO, S., VALENTI, A. (2010). Varieties with at most quadratic growth. ISRAEL JOURNAL OF MATHEMATICS, 178(1), 209-228 [10.1007/s11856-010-0063-4].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10447/47030`