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Subexponential estimations in Shirshov height theorem | | Alexei Belov
; Mikhail Kharitonov
; | | Date: |
25 Jan 2011 | | Abstract: | The paper is devoted to subexponential estimations in Shirshof Height
theorem. Let V be a set of words. It has height h over the set Y (Y is called
Shirshov base) if every word from V has following form v^(k_1)_(i_1) ...
v^(k_s)_(i_s) where v_(i_{alpha}) in Y for all {alpha} and s prec h+1. A
word W is n-divided, if it can be represented in the following form W = W_0 W_1
... W_n such that W_1 prec W_2 prec ... prec W_n. If an affine algebra A
satisfies polynomial identity of degree n then A is spanned by non n-divided
words of generators a_1 prec ... prec a_l. Shirshov proved that the set of
non n-divided words over alphabet of cardinality l has bounded height h over
the set Y consisting of all the words of degree leq n - 1. We show that H
prec {Phi}(n, l), where {Phi}(n, l) = l^(e^2) E_1 n^(E2+9(2e^2+1) ln n), E1
= 576 12^(3(2e^2+1) ln 12) + 1,E2 = 7 + ln 2 + 6(2e^2 + 1) ln 12, e =
2.718281828 . . . - Neper constant. Let l, n и d succ n be positive integers.
Then all the words over alphabet of cardinality l whose length is greater then
{Psi}’ (n, d, l) are either n-divided of contain d-th power of subword, where
{Psi}’(n, d, l) = D_1 l^(e^2) (n^2 d)^((2e 2+1) ln(n^2 d)+D_2) (nd)^4, D_1 =
3^((2e^2+1)(ln 3-1)+2), D_2 = (2e^2 + 1)(2 ln 3 - 1) + 2. This imply
subexponential estimations on the nilpotency index of nill-algebras of an
arbitrary characteristics. Original Shirshov estimation was just recursive, in
1982 double exponent was obtained, in 1992 was obtained exponential estimation.
This answered Zelmanov question in positive way. Our proof uses Latyshev idea
of Dillworth theorem application. | | Source: | arXiv, 1101.4909 | | Services: | Forum | Review | PDF | Favorites | |
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