Quasi-polynomial mappings with constant Jacobian

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Abstract

The famous Jacobian conjecture (JC) remains open even for dimension $2$. In this paper we study it by extending theclass of polynomial mappings to quasi-polynomial ones. We show that any possible non-invertible polynomialmapping with non-zero constant Jacobian can be transformed into a special reduced form by a sequence ofelementary transformations.

About the authors

Sergey Ivanovich Pinchuk

Indiana University

References

  1. S. S. Abhyankar, Lectures on expansion techniques in algebraic geometry, Tata Inst. Fund. Res. Lectures on Math. and Phys., 57, Tata Inst. Fund. Res., Bombay, 1977, iv+168 pp.
  2. H. Bass, E. H. Connel, D. Wright, “The Jacobian conjecture: reduction of degree and formal expansion of the inverse”, Bull. Amer. Math. Soc. (N.S.), 7:2 (1982), 287–330
  3. A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, Progr. Math., 190, Birkhäuser Verlag, Basel, 2000, xviii+329 pp.
  4. E. Formanek, “Theorems of W. W. Stothers and the Jacobian conjecture in two variables”, Proc. Amer. Math. Soc., 139:4 (2011), 1137–1140
  5. C. Valqui, J. A. Guccione, J. J. Guccione, “On the shape of possible counterexamples to the Jacobian conjecture”, J. Algebra, 471 (2017), 13–74
  6. L. Makar-Limanov, On the Newton polygon of a Jacobian mate, MPIM Preprint Series, No. 2013-53, Max-Planck-Institut für Mathematik, Bonn, 2013, 14 pp.
  7. L. Makar-Limanov, On the Newton polytope of a Jacobian pair, MPIM Preprint Series, No. 2014-30, Max-Planck-Institut für Mathematik, Bonn, 2014, 20 pp.
  8. T. T. Moh, “On the Jacobian conjecture and the configurations of roots”, J. Reine Angew. Math., 1983:340 (1983), 140–212
  9. J. Dixmier, “Sur les algèbres de Weyl”, Bull. Soc. Math. France, 96 (1968), 209–242
  10. A. Joseph, “The Weyl algebra – semisimple and nilpotent elements”, Amer. J. Math., 97:3 (1975), 597–615
  11. B. J. Birch, S. Chowla, M. Hall, Jr., A. Schinzel, “On the difference $x^3-y^2$”, Norske Vid. Selsk. Forh. (Trondheim), 38 (1965), 65–69
  12. H. Davenport, “On $f^3(t)-g^2(t)$”, Norske Vid. Selsk. Forh. (Trondheim), 38 (1965), 86–87
  13. W. W. Stothers, “Polynomial identities and Hauptmoduln”, Quart. J. Math. Oxford Ser. (2), 32:127 (1981), 349–370
  14. U. Zannier, “On Davenport's bound for the degree of $f^3-g^2$ and Riemann's existence theorem”, Acta Arith., 71:2 (1995), 107–137
  15. F. Pakovich, A. K. Zvonkin, “Minimum degree of the difference of two polynomials over $mathbb Q$, and weighted plane trees”, Selecta Math. (N.S.), 20:4 (2014), 1003–1065

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