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

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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.

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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].

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We are grateful to Yagzhev’s widow G.I. Yagzheva, and also to Jean-Yves Sharbonel, for providing some unpublished materials.

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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

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  1. Alexei Belov
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  4. Jie-Tai Yu
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Correspondence to Louis Rowen .

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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

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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

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