Skip to main content
SpringerLink
Log in

The Jacobian Conjecture, Together with Specht and Burnside-Type Problems

  • Conference paper
  • First Online:
Automorphisms in Birational and Affine Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 79))

Abstract

We explore an approach to the celebrated Jacobian Conjecture by means of identities of algebras, initiated by the brilliant deceased mathematician, Alexander Vladimirovich Yagzhev (1951–2001), whose works have only been partially published. This approach also indicates some very close connections between mathematical physics, universal algebra, and automorphisms of polynomial algebras.

Dedicated to the memory of A.V. Yagzhev

2010 Mathematics Subject Classification: Primary 13F20, 14E08, 14R15, 17A30, 17A40, 17A50; Secondary 13F25, 17A01, 17A05, 17A15, 17A65.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Algebraic geometers use the word variety, roughly speaking, for objects whose local structure is obtained from the solution of system of algebraic equations. In the framework of universal algebra, this notion is used for subcategories of algebras defined by a given set of identities. A deep analog of these notions is given in [12].

References

  1. A. Abdesselam, The Jacobian Conjecture as a problem of Perturbative Quantum Field theory. Ann. Henri Poincare 4(2), 199–215 (2003)

    MATH  MathSciNet  Google Scholar 

  2. S. Abhyankar, T. Moh, Embedding of the line in the plane. J. Reine Angew. Math. 276, 148–166 (1975)

    MATH  MathSciNet  Google Scholar 

  3. S.A. Amitsur, Algebras over infinite fields. Proc. Am. Math. Soc. 7, 35–48 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  4. S.A. Amitsur, A general theory of radicals: III. Applications. Am. J. Math. 75, 126–136 (1954)

    MathSciNet  Google Scholar 

  5. H. Bass, E.H. Connell, D. Wright, The Jacobian Conjecture: reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc. (N.S.) 7(2), 287–330 (1982)

    Google Scholar 

  6. V.V. Bavula, The inversion formulae for automorphisms of Weil algebras and polynomial algebras. J. Pure Appl. Algebra 210, 147–159 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. V.V. Bavula, The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic. J. Pure Appl. Algebra 212(10), 2320–2337 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. V.V. Bavula, An analogue of the conjecture of Dixmier is true for the algebra of polynomial integro-differential operators. J. Algebra 372, 237–250 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. V.V. Bavula, Every monomorphism of the Lie algebra of unitriangular polynomial derivations is an authomorphism. C. R. Acad. Sci. Paris Ser. 1 350(11–12), 553–556 (2012)

    Google Scholar 

  10. V.V. Bavula, The JacobianConjecture2n implies the Dixmier Problem n . arXiv:math/0512250

    Google Scholar 

  11. A. Beauville, J.-L. Colliot-Thelene, J-J Sansuc, P. Swinnerton-Dyer, Varietes stablement rationnelles non rationnelles. Ann. Math. 121, 283–318 (1985)

    Google Scholar 

  12. A. Belov, Local finite basis property and local representability of varieties of associative rings. Izvestia Russ. Acad. Sci. 74(1), 3–134 (2010). English transl.: Izvestiya Math. 74, 1–126 (2010)

    Google Scholar 

  13. A. Belov, A. Berzins, R. Lipianskii, Automorphisms of the endomorphism group of the free associative algebra. Int. J. Algebra Comput. 17(5/6), 923–939 (2007)

    Article  MATH  Google Scholar 

  14. A. Belov, M.L. Kontsevich, Jacobian and Dixmier Conjectures are stably equivalent. Moscow Math. J. 7(2), 209–218 (2007) (A special volume dedicated to the 60-th anniversary of A.G.Khovanskii)

    Google Scholar 

  15. A. Belov, M.L. Kontsevich, Automorphisms of Weyl algebras. Lett. Math. Phys. 74(3), 181–199 (2005) (A special volume dedicated to the memory of F.A. Berezin)

    Google Scholar 

  16. A. Belov, R. Lipyanskii, Automorphisms of the endomorphism group of the free associative-commutative algebra over an arbitrary field. J. Algebra 333, 40–54 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Belov, L. Makar-Limanov, J.T. Yu, On the generalised cancellation conjecture. J. Algebra 281, 161–166 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Belov, L.H. Rowen, U. Vishne, Structure of Zariski-closed algebras. Trans. Am. Math. Soc. 362, 4695–4734 (2012)

    Article  Google Scholar 

  19. A. Belov, J.-T. Yu, On the lifting of the Nagata automorphism. Selecta Math. (N.S.) 17, 935–945 (2011)

    Google Scholar 

  20. A. Belov-Kanel, J.-T. Yu, Stable tameness of automorphisms of F〈x, y, z〉 fixing z. Selecta Math. (N.S.) 18, 799–802 (2012)

    Google Scholar 

  21. J. Berson, A. van den Essen, D. Wright, Stable tameness of two-dimensional polynomial automorphisms over a regular ring, 2007 (rev. 2010). Adv. Math. 230, 2176–2197 (2012)

    Google Scholar 

  22. J. Birman, An inverse function theorem for free groups. Proc. Am. Math. Soc. 41, 634–638 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  23. P. Bonnet, S. Vénéreau, Relations between the leading terms of a polynomial automorphism. J. Algebra 322(2), 579–599 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. P.M. Cohn, Free Rings and Their Relations, 2nd edn. (Academic, London, 1985)

    MATH  Google Scholar 

  25. A.J. Czerniakiewicz, Automorphisms of a free associative algebra of rank 2, II. Trans. Am. Math. Soc. 171, 309–315 (1972)

    MATH  MathSciNet  Google Scholar 

  26. W. Danielewski, On the cancellation problem and automorphism groups of affine algebraic varieties, Warsaw. Preprint (1989)

    Google Scholar 

  27. W. Dicks, Automorphisms of the free algebra of rank two. Group actions on rings. (Brunswick, Maine, 1984). Contemp. Math. 43, 63–68 (1985)

    Google Scholar 

  28. W. Dicks, J. Levin, Jacobian conjecture for free associative algebras. Commun. Alg. 10(12), 1285–1306 (1982)

    Article  MATH  Google Scholar 

  29. V. Drensky, J.-T. Yu, The strong Anick conjecture. Proc. Natl. Acad. Sci. USA 103, 4836–4840 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. V. Drensky, J.-T. Yu, Coordinates and automorphisms of polynomial and free associative algebras of rank three. Front. Math. China 2(1), 13–46 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. V. Drensky, J.-T. Yu, The strong Anick conjecture is true. J. Eur. Math. Soc. 9, 659–679 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. V. Drensky, J.-T. Yu, A cancellation conjecture for free associative algebras. Proc. Am. Math. Soc. 136, 3391–3394 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. L.M. Druźkowski, The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case. Ann. Polon. Math. 87, 83–92 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. L.M. Druźkowski, New reduction in the Jacobian conjecture. Effective methods in algebraic and analytic geometry, 2000 (Krakw). Univ. Iagel. Acta Math. 39, 203–206 (2001)

    Google Scholar 

  35. A. Van den Essen, Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics, vol. 190 (Birkhauser, Basel, 2000), pp. xviii, 329

    Google Scholar 

  36. A. Van den Essen, The Amazing Image Conjecture. arXiv:1006.5801

    Google Scholar 

  37. A. Van den Essen, M. de Bondt, in Recent Progress on the Jacobian Conjecture. Proc. of the Int. Conf. Singularity Theory in honour of S. Lojawiewicz, Cracow, 22–26 March 2004. Ann. Polon. Math., vol. 87 (2005), pp. 1–11

    Google Scholar 

  38. A. Van den Essen, M. de Bondt, The Jacobian Conjecture for symmetric Druźkowski mappings. Ann. Polon. Math. 86(1), 43–46 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Van den Essen, D. Wright, W. Zhao, On the Image Conjecture. J. Algebra 340, 211—224 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. R.H. Fox, Free differential calculus, I. Derivation in the free group ring. Ann. Math. 57(2), 547–560 (1953)

    Google Scholar 

  41. M.H. Gizatullin, Automorphisms of affine surfaces, I, II. Math. USSR-Izvestiya 11(1), 54–103 (1977)

    MathSciNet  Google Scholar 

  42. G. Gorni, G. Zampieri, Yagzhev polynomial mappings: on the structure of the Taylor expansion of their local inverse. Polon. Math. 64, 285–290 (1996)

    MATH  MathSciNet  Google Scholar 

  43. H.W.E. Jung, Uber ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184, 161–174 (1942)

    MathSciNet  Google Scholar 

  44. S. Kaliman, M. Zaidenberg, Families of affine planes: the existence of a cylinder. Michigan Math. J. 49, 353–367 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  45. E. Kuzmin, I.P. Shestakov, Non-associative structures (English). Algebra VI. Encycl. Math. Sci. 57, 197–280 (1995). Translation from Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. Fundam. Napravleniya 57, 179–266 (1990)

    Google Scholar 

  46. M. Karas’, Multidegrees of tame automorphisms of C n. Dissertationes Math. 477, 55 (2011)

    Google Scholar 

  47. A. Khoroshkin, D. Piontkovski, On generating series of finitely presented operads. Preprint. arXiv:1202.5170 (2012)

    Google Scholar 

  48. V.S. Kulikov, The Jacobian conjecture and nilpotent mappings, in Complex Analysis in Modern Mathematics (FAZIS, Moscow, 2001 in Russian), pp. 167–179. Eng. Trans. J. Math. Sci. 106, 3312–3319 (2001)

    Google Scholar 

  49. W. van der Kulk, On polynomial rings in two variables. Nieuw Arch. Wiskunde 1, 33–41 (1953)

    MATH  MathSciNet  Google Scholar 

  50. S. Kuroda, Shestakov-Umirbaev reductions and Nagata’s conjecture on a polynomial automorphism. Tohoku Math. J. 62, 75–115 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  51. Y.C. Li, J.T. Yu, Degree estimate for subalgebras. J. Algebra 362, 92–98 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  52. L. Makar-Limanov, The automorphisms of the free algebra with two generators. Funkcional. Anal. i Priloen. 4(3), 107–108 (1970 in Russian)

    Google Scholar 

  53. L. Makar-Limanov, A new proof of the Abhyankar-Moh-Suzuki Theorem, 18 pp. arXiv:1212.0163

    Google Scholar 

  54. L. Makar-Limanov, J.-T. Yu, Degree estimate for subalgebras generated by two elements. J. Eur. Math. Soc. 10, 533–541 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  55. M. Markl, S. Shnider, J. Stasheff, Operads in Algebra, Topology and Physics. Math. Surveys and Monographs, vol. 96 (AMS, Providence, 2002)

    Google Scholar 

  56. M. Miyanishi, T. Sugie, Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ. 20, 11–42 (1980)

    MATH  MathSciNet  Google Scholar 

  57. M. Nagata, On the Automorphism Group of k[x,y], Department of Mathematics, Kyoto University, Lectures in Mathematics, vol. 5 (Kinokuniya Book-Store Co., Ltd., Tokyo, 1972), pp. v, 53

    Google Scholar 

  58. J. Nielsen, Die Isomorphismen der allgemeinen, undendlichen Gruppen mit zwei Eerzeugenden. Math. Ann. 78, 385–397 (1918)

    Article  Google Scholar 

  59. J. Nielsen, Die Isomorphismengruppe der freien Gruppen. Math. Ann. 91, 169–209 (1924)

    Article  MATH  MathSciNet  Google Scholar 

  60. A.Yu. Ol’shanskij, Groups of bounded period with subgroups of prime order. Algebra i Logika 21, 553–618 (1982). Translation in Algebra Log. 21, 369–418 (1983)

    Google Scholar 

  61. R. Peretz, Constructing Polynomial Mappings Using Non-commutative Algebras. In Aff. Algebr. Geometry. Contemporary Mathematics, vol. 369 (American Mathematical Society, Providence, 2005), pp. 197–232

    Google Scholar 

  62. D. Piontkovski, Operads versus Varieties: a dictionary of universal algebra. Preprint (2011)

    Google Scholar 

  63. D. Piontkovski, On Kurosh problem in varieties of algebras. Translated from Proceedings of Kurosh conference (Fund. Prikl. Matematika 14, 5, 171–184 (2008)). J. Math. Sci. 163(6), 743–750 (2009)

    Google Scholar 

  64. Yu.P. Razmyslov, Algebras satisfying identity relations of Capelli type. Izv. Akad. Nauk SSSR Ser. Mat. 45, 143–166, 240 (1981 in Russian)

    Google Scholar 

  65. Yu.P. Razmyslov, Identities of Algebras and Their Representations. Sovremennaya Algebra [Modern Algebra] (Nauka, Moscow 1989), p. 432. Translations of Mathematical Monographs, vol. 138 (American Mathematical Society, Providence, 1994), pp. xiv, 318

    Google Scholar 

  66. Yu.P. Razmyslov, K.A. Zubrilin, Nilpotency of obstacles for the representability of algebras that satisfy Capelli identities, and representations of finite type. Uspekhi Mat. Nauk 48, 171–172 (1993 in Russian); Translation in Russ. Math. Surv. 48, 183–184 (1993)

    Google Scholar 

  67. C. Reutenauer, Applications of a noncommutative Jacobian matrix. J. Pure Appl. Algebra 77, 634–638 (1992)

    Article  MathSciNet  Google Scholar 

  68. L.H. Rowen, Graduate algebra: Noncommutative view. Graduate Studies in Mathematics, vol. 91 (AMS, Providence, 2008)

    Google Scholar 

  69. A.H. Schofield, Representations of Rings Over Skew Fields. London Mathematical Society Lecture Note Series, vol. 92 (Cambridge University Press, Cambridge, 1985)

    Google Scholar 

  70. I.R. Shafarevich, On some infinite dimensional groups. Recondidi di Matematica 25, 208–212 (1966)

    MathSciNet  Google Scholar 

  71. I.P. Shestakov, Finite-dimensional algebras with a nil basis. Algebra i Logika 10, 87–99 (1971 in Russian)

    Google Scholar 

  72. I.P. Shestakov, U.U. Umirbaev, Poisson brackets and two-generated subalgebras of rings of polynomials. J. Am. Math. Soc. 17(1), 181–196 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  73. I.P. Shestakov, U.U. Umirbaev, The tame and the wild automorphisms of polynomial rings in three variables. J. Am. Math. Soc. 17(1), 197–227 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  74. V. Shpilrain, On generators of L∕R 2 Lie algebras. Proc. Am. Math. Soc. 119, 1039–1043 (1993)

    MATH  MathSciNet  Google Scholar 

  75. D. Singer, On Catalan trees and the Jacobian conjecture. Electron. J. Combin. 8(1), Research Paper 2, 35 (2001) (electronic)

    Google Scholar 

  76. V. Shpilrain, J.-T. Yu, Affine varieties with equivalent cylinders. J. Algebra 251(1), 295–307 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  77. V. Shpilrain, J.-T. Yu, Factor algebras of free algebras: on a problem of G. Bergman. Bull. Lond. Math. Soc. 35, 706–710 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  78. M. Suzuki, Propiétés topologiques des polynômes de deux variables complexes, et automorphismes algébraique de l’espace C 2. J. Math. Soc. Jpn. 26, 241–257 (1974)

    Article  MATH  Google Scholar 

  79. Y. Tsuchimoto, Preliminaries on Dixmier conjecture. Mem. Fac. Sci. Kochi Univ. Ser. A Math. 24, 43–59 (2003)

    MATH  MathSciNet  Google Scholar 

  80. Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures. Osaka J. Math. 42(2), 435–452 (2005)

    MATH  MathSciNet  Google Scholar 

  81. U.U. Umirbaev, On Jacobian matrices of Lie algebras, in 6th All-Union Conference on Varieties of Algebraic Systems, Magnitogorsk, 1990, pp. 32–33

    Google Scholar 

  82. U.U. Umirbaev, Shreer varieties of algebras. Algebra i Logika 33(3), 317–340, 343 (1994 in Russian). Translation in Algebra Log. 33, 180–193 (1994)

    Google Scholar 

  83. U.U. Umirbaev, Tame and Wild Automorphisms of Polynomial Algebras and Free Associative Algebras (Max-Planck-Institute für Mathematics, Bonn). Preprint MPIM 2004-108

    Google Scholar 

  84. U.U. Umirbaev, On an extension of automorphisms of polynomial rings. Sibirsk. Mat. Zh. 36, 911–916 (1995 in Russian). Translation in Siberian Math. J. 36, 787–791 (1995)

    Google Scholar 

  85. U.U. Umirbaev, The Anick automorphism of free associative algebras. J. Reine Angew. Math. 605, 165–178 (2007)

    MATH  MathSciNet  Google Scholar 

  86. U.U. Umirbaev, Defining relations of the tame automorphism group of polynomial algebras in three variables. J. Reine Angew. Math. 600, 203–235 (2006)

    MATH  MathSciNet  Google Scholar 

  87. U.U. Umirbaev, Defining relations for automorphism groups of free algebras. J. Algebra 314, 209–225 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  88. U.U. Umirbaev, J.T. Yu, The strong Nagata conjecture. Proc. Natl. Acad. Sci. USA 101, 4352–4355 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  89. A.G. Vitushkin, Computation of the Jacobian of a rational transformation of C 2 and some applications. Mat. Zametki 66(2), 308–312 (1999 in Russian). Transl. in Math. Notes 66(1–2), 245–249 (1999)

    Google Scholar 

  90. A.G. Vitushkin, A criterion for the representability of a chain of σ-processes by a composition of triangular chains. Mat. Zametki 65(5), 643–653 (1999 in Russian). Transl. in Math. Notes 65(5–6), 539–547 (1999)

    Google Scholar 

  91. A.G. Vitushkin, Description of the homology of a ramified covering of C 2. Mat. Zametki 64(6), 839–846 (1998 in Russian). Transl. in Math. Notes 64, 726–731 (1999)

    Google Scholar 

  92. J.H.M. Wedderburn, Note on algebras. Ann. Math. 38, 854–856 (1937)

    MathSciNet  Google Scholar 

  93. D. Wright, The Jacobian Conjecture as a Problem in Combinatorics, in the monograph Affine Algebraic Geometry, in honor of Masayoshi Miyanishi,edited by Takayuki Hibi (Osaka University Press, 2007). ArXiv: math.Co/0511214 v2, 22 Mar 2006

    Google Scholar 

  94. D. Wright, The Jacobian Conjecture: ideal membership questions and recent advances. Affine algebraic geometry. Contemp. Math. 369, 261–276 (2005)

    Article  Google Scholar 

  95. A.V. Yagzhev, The generators of the group of tame automorphisms of an algebra of polynomials. Sibirsk. Mat. Ž. 18(1), 222–225, 240 (1977 in Russian)

    Google Scholar 

  96. A.V. Yagzhev, On endomorphisms of free algebras. Sibirsk. Mat. Zh. 21(1), 181–192, 238 (1980 in Russian)

    Google Scholar 

  97. A.V. Yagzhev, On the algorithmic problem of recognition of automorphisms among the endomorphisms of free associate algebras of finite rank. Sibirsk. Mat. Zh. 21(1), 193–199, 238 (1980 in Russian)

    Google Scholar 

  98. A.V. Yagzhev, On a problem of Keller. Sibirsk. Mat. Zh. 21(5), 141–150, 191 (1980 in Russian)

    Google Scholar 

  99. A.V. Yagzhev, Finiteness of the set of conservative polynomials of a given degree. Mat. Zametki 41(2), 148–151, 285 (1987 in Russian)

    Google Scholar 

  100. A.V. Yagzhev, Nilpotency of extensions of an abelian group by an abelian group. Mat. Zametki 43(3), 424–427, 431 (1988 in Russian); translation in Math. Notes 43(3–4), 244–245 (1988)

    Google Scholar 

  101. A.V. Yagzhev, Locally nilpotent subgroups of the holomorph of an abelian group. Mat. Zametki 46(6), 118 (1989 in Russian)

    Google Scholar 

  102. A.V. Yagzhev, A sufficient condition for the algebraicity of an automorphism of a group. Algebra i Logika 28(1), 117–119, 124 (1989 in Russian). Translation in Algebra and Logic 28(1), 83–85 (1989)

    Google Scholar 

  103. A.V. Yagzhev, Invertibility of endomorphisms of free associative algebras. Mat. Zametki 49(4), 142–147, 160 (1991 in Russian). Translation in Math. Notes 49(3–4), 426–430 (1991)

    Google Scholar 

  104. A.V. Yagzhev, Engel algebras satisfying Capelli identities, in Proceedings of Shafarevich Seminar, Moscow, 2000, pp. 83–88 (in Russian)

    Google Scholar 

  105. A.V. Yagzhev, Endomorphisms of polynomial rings and free algebras of different varieties, in Proceedings of Shafarevich Seminar, Moscow, 2000, pp. 15–47 (in Russian)

    Google Scholar 

  106. A. Zaks, Dedekind subrings of K[x 1, …, x n ] are rings of polynomials. Israel J. Math. 9, 285–289 (1971)

    Google Scholar 

  107. E. Zelmanov, On the nilpotence of nilalgebras. Lect. Notes Math. 1352, 227–240 (1988)

    Article  MathSciNet  Google Scholar 

  108. W. Zhao, New Proofs for the Abhyankar-Gurjar Inversion Formula and the Equivalence of the Jacobian Conjecture and the Vanishing Conjecture. Proc. Am. Math. Soc. 139, 3141–3154 (2011)

    Article  MATH  Google Scholar 

  109. W. Zhao, Mathieu subspaces of associative algebras. J. Algebra 350, 245–272 (2012). arXiv:1005.4260

    Google Scholar 

  110. K.A. Zhevlakov, A.M. Slin’ko, I.P. Shestakov, A.I. Shirshov, Nearly Associative Rings (Nauka, Moscow, 1978 in Russian)

    Google Scholar 

  111. K.A. Zubrilin, Algebras that satisfy the Capelli identities. Mat. Sb. 186(3), 53–64 (1995 in Russian). Translation in Sb. Math. 186(3), 359–370 (1995)

    Google Scholar 

  112. K.A. Zubrilin, On the class of nilpotence of obstruction for the representability of algebras satisfying Capelli identities. Fundam. Prikl. Mat. 1(2), 409–430 (1995 in Russian)

    Google Scholar 

  113. K.A. Zubrilin, On the Baer ideal in algebras that satisfy the Capelli identities. Mat. Sb. 189, 73–82 (1998 in Russian). Translation in Sb. Math. 189, 1809–1818 (1998)

    Google Scholar 

Download references

Acknowledgements

The first and third authors are supported by the Israel Science Foundation grant No. 1207/12. The research of Jie-Tai Yu was partially supported by an RGC-GRF Grant.

Yagzhev was a doctoral student of the second author, L. Bokut.

We thank I.P. Shestakov for useful comments, and also thank the referees for many helpful suggestions in improving the exposition.

We are grateful to Yagzhev’s widow G.I. Yagzheva, and also to Jean-Yves Sharbonel, for providing some unpublished materials.

Author information

Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, Ramat Gan, 52900, Israel

    Alexei Belov & Louis Rowen

  2. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Leonid Bokut

  3. South China Normal University, Guangzhou, China

    Leonid Bokut

  4. Department of Mathematics, The University of Hong Kong, Hong Kong SAR, China

    Jie-Tai Yu

Authors
  1. Alexei Belov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonid Bokut
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Louis Rowen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jie-Tai Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Louis Rowen .

Editor information

Editors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom

    Ivan Cheltsov

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Ciro Ciliberto

  3. Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany

    Hubert Flenner

  4. Department of Mathematics, University of California San Diego, La Jolla, California, USA

    James McKernan

  5. Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

    Yuri G. Prokhorov

  6. Institute Fourier, UMR 5582, CNRS-UJF, University of Grenoble I, Saint Martin D’Hères, France

    Mikhail Zaidenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Belov, A., Bokut, L., Rowen, L., Yu, JT. (2014). The Jacobian Conjecture, Together with Specht and Burnside-Type Problems. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_15

Download citation

Publish with us

Policies and ethics

Search

Navigation