Geodesic
Subexponential estimates in Shirshov's theorem on height
Sbornik. Mathematics, Tome 203 (2012) no. 4, pp. 534-553 Cet article a éte moissonné depuis la source Math-Net.Ru

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Suppose that $F_{2,m}$ is a free $2$-generated associative ring with the identity $x^m=0$. In 1993 Zelmanov put the following question: is it true that the nilpotency degree of $F_{2,m}$ has exponential growth? We give the definitive answer to Zelmanov's question by showing that the nilpotency class of an $l$-generated associative algebra with the identity $x^d=0$ is smaller than $\Psi(d,d,l)$, where $$ \Psi(n,d,l)=2^{18}l(nd)^{3\log_3(nd)+13}d^2. $$ This result is a consequence of the following fact based on combinatorics of words. Let $l$, $n$ and $d\geqslant n$ be positive integers. Then all words over an alphabet of cardinality $l$ whose length is not less than $\Psi(n,d,l)$ are either $n$-divisible or contain $x^d$; a word $W$ is $n$-divisible if it can be represented in the form $W=W_0W_1\dotsb W_n$ so that $W_1,\dots,W_n$ are placed in lexicographically decreasing order. Our proof uses Dilworth's theorem (according to V. N. Latyshev's idea). We show that the set of not $n$-divisible words over an alphabet of cardinality $l$ has height $h<\Phi(n,l)$ over the set of words of degree $\le n-1$, where $$ \Phi(n,l)=2^{87}l\cdot n^{12\log_3n+48}. $$ Bibliography: 40 titles.
Keywords: Shirshov's theorem on height, word combinatorics, $n$-divisibility, Dilworth theorem, Burnside-type problems. 
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     title = {Subexponential estimates in {Shirshov's} theorem on height},
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A. Ya. Belov; M. I. Kharitonov. Subexponential estimates in Shirshov's theorem on height. Sbornik. Mathematics, Tome 203 (2012) no. 4, pp. 534-553. http://geodesic.mathdoc.fr/item/SM_2012_203_4_a3/

[1] A. I. Shirshov, “O nekotorykh neassotsiativnykh nil-koltsakh i algebraicheskikh algebrakh”, Matem. sb., 41(83):3 (1957), 381–394 | MR | Zbl

[2] A. I. Shirshov, “O koltsakh s tozhdestvennymi sootnosheniyami”, Matem. sb., 43(85):2 (1957), 277–283 | MR | Zbl

[3] E. I. Zel'manov, “On the nilpotency of nil algebras”, Algebra – some current trends (Varna, 1986), Lecture Notes in Math., 1352, Springer-Verlag, Berlin, 1988, 227–240 | DOI | MR | Zbl

[4] A. Ya. Belov, V. V. Borisenko, V. N. Latyshev, “Monomial algebras”, J. Math. Sci. (New York), 87:3 (1997), 3463–3575 | DOI | MR | Zbl

[5] A. R. Kemer, selected papers of A. I. Shirshov

[6] A. Kanel-Belov, L. H. Rowen, selected papers of A. I. Shirshov

[7] V. A. Ufnarovskii, “Kombinatornye i asimptoticheskie metody v algebre”, Algebra – 6, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Fundam. napravleniya, 57, VINITI, M., 1990, 5–177 | MR | MR | Zbl | Zbl

[8] V. Drensky, E. Formanek, Polynomial identity rings, Adv. Courses Math. CRM Barcelona, Birkhäuser, Basel, 2004 | MR | Zbl

[9] S. V. Pchelintsev, “A theorem on height for alternative algebras”, Math. USSR-Sb., 52:2 (1985), 541–551 | DOI | MR | Zbl | Zbl

[10] S. P. Mishchenko, “A variant of the height theorem for Lie algebras”, Math. Notes, 47:4 (1990), 368–372 | DOI | MR | Zbl | Zbl

[11] A. Ya. Belov, “On a Shirshov basis of relatively free algebras of complexity $n$”, Math. USSR-Sb., 63:2 (1989), 363–374 | DOI | MR | Zbl | Zbl

[12] A. Kanel-Belov, L. H. Rowen, Computational aspects of polynomial identities, Res. Notes in Math., 9, Peters, Wellesley, MA, 2005 | MR | Zbl

[13] Gh. Ciocanu, “Independence and quasiregularity in algebras. II”, Izv. Akad. Nauk Respub. Moldova Mat., 1 (1997), 70–77 | MR

[14] G. P. Chekanu, “O lokalnoi konechnosti algebr”, Matem. issled., 105 (1988), 153–171 | MR | Zbl

[15] G. P. Chekanu, E. P. Kozukhar, “Nezavisimost i nilpotentnost v algebrakh”, Izv. AN Moldovy. Matem., 2 (1993), 51–62 | MR | Zbl

[16] G. P. Chekanu, “Independence and quasiregularity in algebras”, Russian Acad. Sci. Dokl. Math., 50:1 (1995), 84–89 | MR | Zbl

[17] V. A. Ufnarovskiǐ, “An independence theorem and its consequences”, Math. USSR-Sb., 56:1 (1987), 121–129 | DOI | MR | Zbl | Zbl

[18] V. A. Ufnarovskii, G. P. Chekanu, “Na nilpotentnykh matritsakh”, Matem. issled., 85 (1985), 130–141 | MR | Zbl

[19] A. Ya. Belov, “On the rationality of Hilbert series of relatively free algebras”, Russian Math. Surveys, 52:2 (1997), 394–395 | DOI | MR | Zbl

[20] J. Berstel, D. Perrin, “The origins of combinatorics on words”, European J. Combin., 28:3 (2007), 996–1022 | DOI | MR | Zbl

[21] M. Lothaire, Combinatorics of words (Waterloo, ON, Canada, 1982), Encyclopedia Math. Appl., 17, Addison-Wesley, Reading, MA, 1983 | MR | Zbl

[22] M. Lothaire, Algebraic combinatorics on words, Encyclopedia Math. Appl., 90, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[23] V. N. Latyshev, “Combinatorial generators of the multilinear polynomial identities”, J. Math. Sci., 149:2 (2008), 1107–1112 | DOI | MR | Zbl

[24] A. G. Kolotov, “O verkhnei otsenke vysoty v konechno porozhdennykh algebrakh s tozhdestvami”, Sib. matem. zhurn., 23:1 (1982), 187–189 | MR | Zbl

[25] A. Ya. Belov, “Some estimations for nilpotence of nill-algebras over a field of an arbitrary characteristic and height theorem”, Comm. Algebra, 20:10 (1992), 2919–2922 | DOI | MR | Zbl

[26] V. Drensky, Free algebras and PI-algebras. Graduate course in algebra, Springer-Verlag, Singapore, 2000 | MR | Zbl

[27] M. I. Kharitonov, “Otsenki na strukturu kusochnoi periodichnosti v teoreme Shirshova o vysote”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., V pechati

[28] A. A. Klein, “Indices of nilpotency in a PI-ring”, Arch. Math. (Basel), 44:4 (1985), 323–329 | DOI | MR | Zbl

[29] A. A. Klein, “Bounds for indices of nilpotency and nility”, Arch. Math. (Basel), 74:1 (2000), 6–10 | DOI | MR | Zbl

[30] E. S. Chibrikov, “O vysote Shirshova konechnoporozhdennoi assotsiativnoi algebry, udovletvoryayuschei tozhdestvu stepeni chetyre”, Izv. Altaiskogo gos. un-ta, 1 (2001), 52–56 | Zbl

[31] M. I. Kharitonov, “Dvustoronnie otsenki suschestvennoi vysoty v teoreme Shirshova o vysote”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2012, no. 2, 24–28

[32] M. Kharitonov, Estimations of the particular periodicity in case of the extremal periods in Shirshov's height theorem, arXiv: 1108.6295

[33] A. A. Lopatin, On the nilpotency degree of the algebra with identity $x^n=0$, arXiv: 1106.0950

[34] Dnestrovskaya tetrad. Nereshennye problemy teorii kolets i modulei, 4-e izd., Izd. in-ta matem. SO AN SSSR, Novosibirsk, 1993, 73 | MR | Zbl

[35] I. I. Bogdanov, “Teorema Nagaty–Khigmana dlya polukolets”, Fundament. i prikl. matem., 7:3 (2001), 651–658 | MR | Zbl

[36] C. Procesi, Rings with polynomial identities, Marcel Dekker, New York, 1973 | MR | Zbl

[37] A. Ya. Belov, “The Gel'fand–Kirillov dimension of relatively free associative algebras”, Sb. Math., 195:12 (2004), 1703–1726 | DOI | MR | Zbl

[38] E. N. Kuzmin, “O teoreme Nagaty–Khigmana”, Sbornik trudov, posvyaschennyi 60-letiyu akad. Ilieva, Sofiya, 1975, 101–107

[39] Yu. P. Razmyslov, Identities of algebras and their representations, Transl. Math. Monogr., 138, Amer. Math. Soc., Providence, RI, 1992 | MR | MR | Zbl | Zbl

[40] A. Ya. Belov, “Burnside-type problems, theorems on height, and independence”, J. Math. Sci., 156:2 (2009), 219–260 | DOI | MR | Zbl