Polynomial Automorphisms and the Jacobian Conjecture

In this paper we give an update survey of the most important results concerning the Jacobian conjecture: several equivalent descriptions are given and various related conjectures are discussed. At the end of the paper, we discuss the recent counter-examples, in all dimensions greater than two, to the Markus-Yamabe conjecture (Global asymptotic Jacobian conjecture). Resume Dans ce papier nous presentons un rapport actualise sur les resultats les plus importants concernant la conjecture Jacobienne : plusieurs formulations equivalentes et diverses conjectures connexes sont considerees. A la fin du papier, nous donnons les contre-exemples recents, en toute dimension plus grande que deux, a la conjecture de Markus-Yamabe.

[1]  Khuram Hussain.  Press , 2019, Keywords in Radical Philosophy and Education.

[2]  S. Oda.  Endomorphisms of polynomial rings and Jacobians , 2004, math/0402051.

[3]  Engelbert Hubbers,et al.  A polynomial counterexample to the Markus-Yamabe conjecture , 1997 .

[4]  E.M.G.M. Hubbers,et al.  Polynomial Maps with strongly nilpotent Jacobian matrix and the Jacobian Conjecture , 1996 .

[5]  G. Gorni,et al.  On the Existence of Global Analytic Conjugations for Polynomial Mappings of Yagzhev Type , 1996 .

[6]  B. Deng,et al.  Conjugation for polynomial mappings , 1995 .

[7]  J. Strelcyn,et al.  Generators of rings of constants for some diagonal derivations in polynomial rings , 1995 .

[8]  Jie-Tai Yu,et al.  On the Jacobian Conjecture: Reduction of Coefficients , 1995 .

[9]  S. Maslamani,et al.  Some results on the Jacobian conjecture in higher dimension , 1994 .

[10]  J. Gwozdziewicz.  Injectivity on one line , 1993, alg-geom/9305008.

[11]  Ludwik M. Drużkowski.  The Jacobian Conjecture in case of rank or corank less than three , 1993 .

[12]  D. Wright.  The jacobian conjecture: linear triangularizatlon for cubics in dimension three , 1993 .

[13]  H. Derksen.  The kernel of a derivation , 1993 .

[14]  M. Kang.  A note on the birational cancellation problem , 1992 .

[15]  P. Roberts.  An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert's fourteenth problem , 1990 .

[16]  M. Kang.  The biregular cancellation problem , 1987 .

[17]  J.-J. Sansuc,et al.  Varietes stablement rationnelles non rationnelles , 1985 .

[18]  Ludwik M. Drużkowski.  An effective approach to Keller's Jacobian Conjecture , 1983 .

[19]  Pekka Nousiainen,et al.  Automorphisms of polynomial and power series rings , 1983 .

[20]  H. Bass,et al.  The Jacobian conjecture: Reduction of degree and formal expansion of the inverse , 1982 .

[21]  S. Wang.  A Jacobian criterion for separability , 1980 .

[22]  T. Kambayashi.  Automorphism group of a polynomial ring and algebraic group action on an affine space , 1979 .

[23]  S. Abhyankar,et al.  On the uniqueness of the coefficient ring in a polynomial ring , 1972 .

[24]  M. Rosenlicht,et al.  Injective morphisms of real algebraic varieties , 1962 .

[25]  A. Magnus.  On polynomial Solutions of a differential equation , 1955 .

[26]  O. Keller,et al.  Ganze Cremona-Transformationen , 1939 .

[27]  Victoria Powers.  SOCIÉTÉ MATHÉMATIQUE DE FRANCE , 2018 .

[28]  Engelbert Hubbers,et al.  Chaotic polynomial automorphisms: Counterexamples to several conjectures , 1995 .

[29]  A. Essen.  A Counterexample to a Conjecture of Meisters , 1995 .

[30]  A. Essen.  Seven Lectures on Polynomial Automorphisms , 1995 .

[31]  C. Cheng,et al.  Radial Similarity of Newton Polygons , 1995 .

[32]  G. Meisters.  Polyomorphisms Conjugate to Dilations , 1995 .

[33]  H. Kraft.  Challenging problems on affine $n$-space , 1995 .

[34]  Toru Sugie.  Algebraic Characterization of the Affine Plane and the Affine 3-Space , 1989 .

[35]  Gerald W. Schwarz,et al.  Topological Methods in Algebraic Transformation Groups , 1989 .

[36]  M. Nagata.  Two-dimensional Jacobian conjecture , 1989 .

[37]  H. Kraft.  Algebraic Automorphisms of Affine Space , 1989 .

[38]  M. Nagata,et al.  Rings of constants for $k$-derivations in $k[x_1, \ldots, x_n]$ , 1988 .

[39]  N. Barabanov,et al.  On the Kalman problem , 1988 .

[40]  Lee Rudolph,et al.  Embeddings of the line in the plane. , 1982 .

[41]  A. V. Yagzhev.  Keller's problem , 1980 .

[42]  M. Miyanishi,et al.  Affine surfaces containing cylinderlike open sets , 1980 .

[43]  T. Fujita.  On Zariski problem , 1979 .

[44]  S. Abhyankar,et al.  Embeddings of the line in the plane. , 1975 .

[45]  J. Dixmier.  Sur les algèbres de Weyl , 1968 .

[46]  Masayoshi Nagata,et al.  Lectures on the fourteenth problem of Hilbert , 1965 .

[47]  Lawrence Markus,et al.  Global stability criteria for differential systems. , 1960 .