Journal of Mathematical Sciences

, Volume 193, Issue 4, pp 493–515 | Cite as

Subexponential estimates in the height theorem and estimates on numbers of periodic parts of small periods

  • A. Ya. BelovEmail author
  • M. I. Kharitonov


This paper is devoted to subexponential estimates in Shirshov’s height theorem. A word W is n-divisible if it can be represented in the form W = W 0 W 1 ⋯W n , where W 1W 2W n . If an affine algebra A satisfies a polynomial identity of degree n, then A is spanned by non-n-divisible words of generators a 1a l . A. I. Shirshov proved that the set of non-n-divisible words over an alphabet of cardinality l has bounded height h over the set Y consisting of all words of degree ≤ n − 1. We show that h < Φ (n, l), where Φ(n, l) = 287 ln 12 log3 n+48. Let l, n, and dn be positive integers. Then all words over an alphabet of cardinality l whose length is greater than ψ(n, d, l) are either n-divisible or contain the dth power of a subword, where ψ(n, d, l) = 218 l(nd)3 log3(nd)+13 d 2. In 1993, E. I. Zelmanov asked the following question in the Dniester Notebook: Suppose that F 2,m is a 2-generated associative ring with the identity x m = 0. Is it true that the nilpotency degree of F 2,m has exponential growth? We give the definitive answer to E. I. Zelmanov by this result. We show that the nilpotency degree of the l-generated associative algebra with the identity x d = 0 is smaller than ψ(d, d, l). This implies subexponential estimates on the nilpotency index of nil-algebras of arbitrary characteristic. Shirshov’s original estimate was just recursive; in 1982 a double exponent was obtained, and an exponential estimate was obtained in 1992. Our proof uses Latyshev’s idea of an application of the Dilworth theorem. We think that Shirshov’s height theorem is deeply connected to problems of modern combinatorics. In particular, this theorem is related to the Ramsey theory. We obtain lower and upper estimates of the number of periods of length 2, 3, n − 1 in some non-n-divisible word. These estimates differ only by a constant.


Equivalence Class Polynomial Identity Free Algebra Hilbert Series Exponential Estimate 
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  1. 1.
    A. Ya. Belov, “On a Shirshov basis for relatively free algebras of complexity n,” Mat. Sb., 135, No. 31, 373–384 (1988).Google Scholar
  2. 2.
    A. Ya. Belov, “Some estimations for nilpotency of nil-algebras over a field of an arbitrary characteristic and height theorem,” Commun. Algebra, 20, No. 10, 2919–2922 (1992).zbMATHCrossRefGoogle Scholar
  3. 3.
    A. Ya. Belov, “Rationality of Hilbert series for relatively free algebras,” Usp. Mat. Nauk, 52, No. 2, 153–154 (1997).CrossRefGoogle Scholar
  4. 4.
    A. Ya. Belov, “Gelfand–Kirillov dimension of relatively free associative algebras,” Mat. Sb., 195, No. 12, 3–26 (2004).CrossRefGoogle Scholar
  5. 5.
    A. Ya. Belov, “Burnside-type problems, height and independence theorems,” Fundam. Prikl. Mat., 13, No. 5, 19–79 (2007).Google Scholar
  6. 6.
    A. Ya. Belov, V. V. Borisenko, and V. N. Latyshev, “Monomial algebras,” J. Math. Sci., 87, No. 3, 3463–3575 (1997).MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    A. Ya. Belov and M. I. Kharitonov, “Subexponential estimates in Shirshov theorem on height,” Mat. Sb., 203, No. 4, 81–102 (2012), arXiv:1101.4909.MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Ya. Belov and L. H. Rowen, Computational Aspects of Polynomial Identities, Research Notes Math., Vol. 9, Peters, Wellesley (2005).Google Scholar
  9. 9.
    J. Berstel and D. Perrin, “The origins of combinatorics on words,” Eur. J. Combin., 28, 996–1022 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    I. I. Bogdanov, “Nagata–Higman theorem for semirings,” Fundam. Prikl. Mat., 7, No. 3, 651–658 (2001).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gh. Chekanu, “Local finiteness of algebras,” Mat. Issled., 105, 153–171, 198 (1988).MathSciNetzbMATHGoogle Scholar
  12. 12.
    G. P. Chekanu, “Independence and quasiregularity in algebras,” Dokl. Akad. Nauk, 337, No. 3 (1994).Google Scholar
  13. 13.
    G. P. Chekanu and E. P. Kozhukhar, “Independence and nilpotency in algebras,” Izv. Akad. Nauk Respub. Moldova Mat., No. 2, 51–62, 92–93, 95 (1993).Google Scholar
  14. 14.
    Ye. S. Chibrikov, “On Shirshov height of a finitely generated associative algebra satisfying an identity of degree four,” Izv. Altaisk. Gos. Univ., No. 1 (19), 52–56 (2001).Google Scholar
  15. 15.
    Gh. Ciocanu, “Independence and quasiregularity in algebras. II,” Izv. Akad. Nauk Respub. Moldova Mat., No. 70, 70–77, 132, 134 (1997).Google Scholar
  16. 16.
    Dnestr Copy-Book: A Collection of Operative Information [in Russian], Inst. Mat. SO AN SSSR, Novosibirsk (1993).Google Scholar
  17. 17.
    V. Drensky, Free Algebras and PI-Algebras: Graduate Course in Algebra, Springer, Singapore (2000).zbMATHGoogle Scholar
  18. 18.
    V. Drensky and E. Formanek, Polynomial Identity Ring, Adv. Courses Math., Birkhäuser, Basel (2004).Google Scholar
  19. 19.
    A. Kanel-Belov and L. H. Rowen, “Perspectives on Shirshov’s height theorem,” in: Selected Papers of A. I. Shirshov, Birkhäuser, Basel (2009), pp. 3–20.Google Scholar
  20. 20.
    A. R. Kemer, “Comments on Shirshov’s height theorem,” in: Selected Papers of A. I. Shirshov, Birkhäuser, Basel (2009), pp. 41–48.Google Scholar
  21. 21.
    M. Kharitonov, “Estimations of the particular periodicity in case of the small periods in Shirshov height theorem,” arXiv:1108.6295.Google Scholar
  22. 22.
    M. I. Kharitonov. “Two-sided estimates for essential height in Shirshov height theorem,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 24–28 (2012).Google Scholar
  23. 23.
    M. I. Kharitonov, “Estimates for structure of piecewise periodicity in Shirshov height theorem,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., to appear.Google Scholar
  24. 24.
    A. A. Klein, “Indices of nilpotency in a PI-ring,” Arch. Math., 44, No. 4, 323–329 (1985).MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    A. A. Klein, “Bounds for indices of nilpotency and nility,” Arch. Math., 74, No. 1, 6–10 (2000).MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    A. G. Kolotov, “An upper estimate for height in finitely generated algebras with identities,” Sib. Mat. Zh., 23, No. 1, 187–189 (1982).MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ye. N. Kuzmin, “On Nagata–Higman theorem,” in: Collection of Papers Dedicated to 60th Anniversary of Acad. Iliev, Sofia (1975), pp. 101–107.Google Scholar
  28. 28.
    V. N. Latyshev, “On Regev’s theorem on indentities in a tensor product of PI-algebras,” Usp. Mat. Nauk, 27, 213–214 (1972).zbMATHGoogle Scholar
  29. 29.
    V. N. Latyshev, “Combinatorial generators of polylinear polynomial identities,” Fundam. Prikl. Mat., 12, No. 2, 101–110 (2006).MathSciNetGoogle Scholar
  30. 30.
    A. A. Lopatin, On the Nilpotency Degree of the Algebra with Identity x n = 0, arXiv:1106.0950v1.Google Scholar
  31. 31.
    M. Lothaire, Algebraic Combinatorics on Words, Cambridge Univ. Press, Cambridge (2002).zbMATHCrossRefGoogle Scholar
  32. 32.
    S. P. Mishchenko, “A variant of the height theorem for Lie algebras,” Mat. Zametki, 47, No. 4, 83–89 (1990).MathSciNetzbMATHGoogle Scholar
  33. 33.
    S. V. Pchelintsev, “The height theorem for alternative algebras,” Mat. Sb., 124, No. 4, 557–567 (1984).MathSciNetGoogle Scholar
  34. 34.
    C. Procesi, Rings with Polynomial Identities, New York (1973).Google Scholar
  35. 35.
    Yu. P. Razmyslov, Identities of Algebras and Their Representations [in Russian], Nauka, Moscow (1989).Google Scholar
  36. 36.
    A. I. Shirshov, “On rings with identical relations,” Mat. Sb., 43, No. 2, 277–283 (1957).MathSciNetGoogle Scholar
  37. 37.
    A. I. Shirshov, “On some nonassociative nilrings and algebraic algebras,” Mat. Sb., 41, No. 3, 381–394 (1957).MathSciNetGoogle Scholar
  38. 38.
    V. A. Ufnarovsky, “An independence theorem and its consequences,” Mat. Sb., 128 (170), No. 1 (9), 124–132 (1985).Google Scholar
  39. 39.
    V. A. Ufnarovsky, “Combinatorial and asymptotical methods in algebra,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fund. Napr., 57, 5–177 (1990).Google Scholar
  40. 40.
    V. A. Ufnarovskii and G. P. Chekanu, “Nilpotent matrices,” Mat. Issled., 85, 130–141, 155 (1985).MathSciNetGoogle Scholar
  41. 41.
    E. Zelmanov, “On the nilpotency of nil algebras,” in: L. L. Avramov and K. B. Tchakirian, eds., Proc. 5th Nat. School in Algebra Held in Varna, Bulgaria, Sept. 24 – Oct. 4, 1986, Lect. Notes Math., Vol. 1352, Springer, Berlin (1988), pp. 227–240.Google Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Moscow Institute of Open EducationMoscowRussia
  2. 2.Jacobs University BremenBremenGermany
  3. 3.Moscow State UniversityMoscowRussia

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