Skip to main content
SpringerLink
Log in

Mathieu-Zhao Spaces and the Jacobian Conjecture

  • Conference paper
  • First Online:
Book cover Polynomial Rings and Affine Algebraic Geometry (PRAAG 2018)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 319))

  • 553 Accesses

Abstract

In this paper we define the notion of a Mathieu-Zhao space, give various examples of this concept and use the framework of these Mathieu-Zhao spaces to describe a chain of challenging conjectures, all implying the Jacobian Conjecture.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
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

References

  1. Bass, H., Connell, E., Wright, D.: The jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc.(NS) 7, 287–330. [MR 83k:14028]

    Google Scholar 

  2. de Bondt, M.: A few remarks on the generalized vanishing conjecture. Arch. Math (Basel) 100(6), 533–538 (2013)

    Article  MathSciNet  Google Scholar 

  3. de Bondt, M., van den Essen, A.: A reduction of the jacobian conjecture to the symmetric case. Proc. Am. Math. Soc. 133(8), 2201–2205 (2005). [MR2138860]

    Google Scholar 

  4. Derksen, H., van den Essen, A., Zhao, W.: The gaussian moments conjecture and the jacobian conjecture. Israel J. of Math. 219, 917–928 (2017). arxiv.org/pdf/1506.05192.pdf

  5. Duistermaat, J., van der Kallen, W.: Constant terms in powers of a Laurent polynomial. Indag. Math. (N.S.) 9(2), 221–231 (1998)

    Google Scholar 

  6. van den Essen, A., Wright, D., Zhao, W.: On the image conjecture. J. Alg. 340(1), 211–224 (2011). See also arXiv:1008.3962 [math.RA]

  7. van den Essen, A.: An introduction to Mathieu subspaces. Lectures delivered at the Chern Institute of Mathematics. Tianjin, China (July 2014)

    Google Scholar 

  8. van den Essen, A., Zhao, W.: Two results on homogeneous Hessian nilpotent polynomials. J. Pure Appl. Algebra 212(10), 2190–2193 (2008). See also http://arxiv.org/abs/0704.1690v1

  9. Francoise, J.P., Pakovich, F., Yomdin, Y., Zhao, W.: Moment vanishing problem and positivity: some examples. Bull. des Sci. Math. 135(1), 10–32 (2011). Also available at:http://www.math.bgu.ac.ilakovich/Publications/FPYZ.pdf

  10. Keller, O.H.: Ganze Cremona-Transformationen. Monatsh. Math. Phys. 47, 299–306 (1939)

    Article  MathSciNet  Google Scholar 

  11. Mathieu, O.: Some conjectures about invariant theory and their applications. Algèbre Non Commutative, Groupes Quantiques et Invariants (Reims, 1995). Séminaries et Congrès, vol. 2, pp. 263–279 Société Mathématique de France (1997)

    Google Scholar 

  12. Yagzhev, A.V.: On Keller’s problem. Siberian Math. J. 21, 747–754 (1980)

    Article  MathSciNet  Google Scholar 

  13. Zhao, W.: Hessian nilpotent polynomials and the Jacobian conjecture. Trans. Am. Math. Soc. 359(1), 294–274 (2007). See also math.CV/0409534

    Google Scholar 

  14. Zhao, W.: A vanishing conjecture on differential operators with constant coefficients. Acta Math. Vietnam. 32(3), 259–286 (2007). See also arXiv:0704.1691v2

  15. Zhao, W.: Images of commuting differential operators of order one with constant leading coefficients. J. Algebra 324(2), 231–247 (2010). See also arXiv:0902.0210 [math.CV]

  16. Zhao, W.: Generalizations of the image conjecture and the Mathieu conjecture. J. Pure Appl. Algebra 214, 1200–1216 (2010). See also arXiv:0902.0212 [math.CV]

  17. Zhao, W.: New proofs for the Abhyankar-Gujar inversion formula and equivalence of the Jacobian conjecture and the vanishing conjecture. Proc. AMS 139,(9), 3141–3154 (2011)

    Google Scholar 

  18. Zhao, W.: Mathieu subspaces of associative algebras. J. Algebra 350(1), 245–272 (2012). See also arXiv:1005.4260

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics Astrophysics and Particle Physics, Radboud University Nijmegen, Nijmegen, The Netherlands

    Arno van den Essen

Authors
  1. Arno van den Essen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arno van den Essen .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, Tokyo Metropolitan University, Hachioji, Tokyo, Japan

    Shigeru Kuroda

  2. Faculty of Applied Physics, University of Fukui, Fukui, Japan

    Nobuharu Onoda

  3. Department of Mathematics, Western Michigan University, Kalamazoo, MI, USA

    Gene Freudenburg

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

van den Essen, A. (2020). Mathieu-Zhao Spaces and the Jacobian Conjecture. In: Kuroda, S., Onoda, N., Freudenburg, G. (eds) Polynomial Rings and Affine Algebraic Geometry. PRAAG 2018. Springer Proceedings in Mathematics & Statistics, vol 319. Springer, Cham. https://doi.org/10.1007/978-3-030-42136-6_14

Download citation

Publish with us

Policies and ethics

Search

Navigation