Geodesic
Piecewise periodicity structure estimates in Shirshov's height theorem
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2013), pp. 10-16

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The Gelfand–Kirillov dimension of $l$-generated general matrixes is $(l-1)n^2+1.$ The minimal degree of the identity of this algebra is $2n$ as a corollary of Amitzur–Levitsky theorem. That is why the essential height of $A$ being an $l$-generated PI-algebra of degree $n$ over every set of words can be greater than $(l-1)n^2/4 + 1.$ We prove that if $A$ has a finite GK-dimension, then the number of lexicographically comparable subwords with the period $(n-1)$ in each monoid of $A$ is not greater than $(l-2)(n-1).$ The case of the subwords with the period $2$ is generalized to the proof of Shirshov's Height theorem. 
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     author = {M. I. Kharitonov},
     title = {Piecewise periodicity structure estimates in {Shirshov's} height theorem},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {10--16},
     publisher = {mathdoc},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2013_1_a1/}
}
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M. I. Kharitonov. Piecewise periodicity structure estimates in Shirshov's height theorem. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2013), pp. 10-16. http://geodesic.mathdoc.fr/item/VMUMM_2013_1_a1/