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An independence theorem and its consequences
Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 121-129 Cet article a éte moissonné depuis la source Math-Net.Ru

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The following theorem is proved: Let $A_1,\dots,A_d$ be linear operators in a vector space $V$, $v\in V$, and let the word $C=A_{k_1}A_{k_2}\dots A_{k_n}$ be maximal in the right lexicographical order among all words of length $n$ satisfying the condition $Cv\ne0$. If all the operators corresponding to the subwords of $C$ are nilpotent, then the vectors $v$, $A_{k_n}v$, $A_{k_{n-1}}A_{k_n}v,\dots,A_{k_1}A_{k_2}\cdots A_{k_n}v$ are independent. As a corollary, a proof is presented of Shestakov's conjecture about the number of nil-conditions necessary for a subalgebra of a matrix algebra to be nilpotent. Bibliography: 5 titles. 
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V. A. Ufnarovskii. An independence theorem and its consequences. Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 121-129. http://geodesic.mathdoc.fr/item/SM_1987_56_1_a7/

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