Bulletin (New Series) of the American Mathematical Society

The Jacobian conjecture: Reduction of degree and formal expansion of the inverse

Hyman Bass, Edwin H. Connell, and David Wright

Full-text: Open access

Article information

Bull. Amer. Math. Soc. (N.S.) Volume 7, Number 2 (1982), 287-330.

First available in Project Euclid: 4 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10] 13B10: Morphisms 13B15 14E05: Rational and birational maps 14E07: Birational automorphisms, Cremona group and generalizations
Secondary: 05C05: Trees


Bass, Hyman; Connell, Edwin H.; Wright, David. The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 287--330.https://projecteuclid.org/euclid.bams/1183549636

Export citation


  • [A + K] A. Altman and S. Kleiman, Introduction to Grothendieck duality, Lecture Notes in Math., vol. 146, Springer-Verlag, Berlin and New York, 1970.
  • [Abl] S. S. Abhyankar, Expansion techniques in algebraic geometry, Tata Inst. Fundamental Research, Bombay, 1977.
  • [Ab2] S. S. Abhyankar, Lectures in algebraic geometry, Notes by Chris Christensen, Purdue Univ., 1974.
  • [Ab3] S. S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), 409-448.
  • [A + H + E] S. S. Abhyankar, W. Heinzer and P. Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310-342.
  • [A + M] S. S. Abhyankar and T.-T. Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 149-166.
  • [B] H. Bass, Algebraic K-theory, Benjamin, New York, 1968.
  • [B + M) S. Bochner and W. T. Martin, Several complex variables, Princeton Univ. Press, Princeton, N. J., 1948.
  • [C] L. A. Campbell, A condition for a polynomial map to be invertible, Math. Ann. 205 (1973), 243-248.
  • [C + L] I. Canals and E. Lluis, Acerca de un resultado de Segre, Anal. Inst. Matematicas, Univ. Nacional Autonoma de Mexico 10 (1970), 1-15.
  • [Co] E. H. Connell, A K-theory for the category of projective algebras, J. Pure Appl. Algebra 5 (1974), 281-292.
  • [E] W. Engel, Ein Satz über ganze Cremona Transformationen der Ebene, Math. Ann. 130 (1955), 11-19.
  • [F] L. Fridman, On a characterization of polynomial endomorphisms of Cn, Math. USSR Izv. 7 (1973), 319-328. (Transi, from Izv, Akad. Nauk SSR 37 (1973).)
  • [Fu] T. Fujita, On Zariski problem, Proc. Japan Acad. 55 (1979), 106-110.
  • [G] I. J. Good, Generalization to several variables of Lagrange's expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56 (1960), 367-380.
  • [Gr] W. Gröbner, Sopra un teorema di B. Segre, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. 31 (1961), 118-122. MR 24 (1962) A3155 (I. Barsotti).
  • [J] C. G. J. Jacobi, De resolutione aequationum per series infinitas, J. Reine Angew. Math. 6 (1830), 257-286.
  • [Ja] A. V. Jagžev, On Keller's problem, Siberian Math. J. 21 (1980), 141-150. (Russian)
  • [Jo] S. A. Joni, Lagrange inversion in higher dimension and umbral operators, Linear and Multilinear Algebra 6 ( 1978), 111 -121.
  • [Ju] H. W. E. Jung, Über ganze birationale Transformation der Ebene, J. Reine Angew. Math. 184 (1942), 161-174.
  • [K] O. H. Keller, Ganze Cremona-Transformationen, Monats. Math. Physik 47 (1939), 299-306.
  • [Ma] A. Magnus, On polynomial solutions of a differential equation, Math. Scand. 3 (1955), 255-260.
  • [M-L] G. Makar-Limanov, Automorphisms of a free algebra on two generators, Functional Anal. Appl. 4 (1970), 262-264 (Transl. from Funkcional Anal. i. Prilozen 4 (1970), 107-108.)
  • [Mat] H. Matsumura, Commutative algebra, Benjamin, New York, 1970.
  • [Me] Gary H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math, (to appear).
  • [Mo] T.-T. Moh, On the Jacobian Conjecture and the configuration of roots, J. Reine Angew. Math, (to appear).
  • [Mu] J. P. Murre, An introduction to Grothendieck's theory of the fundamental group, Tata Inst. Fundamental Research, Bombay, 1967.
  • [N + B] Y. Nakai and K. Baba, A generalization of Magnus' theorem, Osaka J. Math. 14 (1977), 403-409.
  • [N1] Pekka Nousiainen, On the Jacobian problem in positive characteristic, Pennsylvania State Univ., preprint, 1981.
  • [N2] Pekka Nousiainen, On the degrees of smooth maps of affine space, Pennsylvania State Univ., preprint, 1981.
  • [N + S] P. Nousiainen and M. Sweedler, Automorphisms of polynomial and power series rings, Cornell Univ., preprint, 1981.
  • [O] Susumu Oda, The Jacobian problem and the simply-connectedness of An over a field k of characteristic zero, Osaka Univ., preprint, 1980.
  • [R] M. Razar, Polynomial maps with constant Jacobian, Israel J. Math. 32 (1979), 97-106.
  • [S] I. R. Shafarevich, On some infinite dimensional groups, Simposio Internazionale di Geometria Algebrica, 1965, published in Rendiconti di Matematica e delle sue applicazioni, Ser. 5, vol. 25, 1966, pp. 208-2121.
  • [Se1] B. Segre, Corrispondenze di Mobius e Transformazioni cremoniane intere, Atti della Accademia delle Scienze di Torino, Classe di Scienzo Fisiche, Mat. e Natural. 91 (1956-57), 3-19.
  • [Se2]B. Segre, Forme differenziali e loro integrali, vol. II, Docet, Roma, 1956.
  • [Se3] B. Segre, Variazione continua ed omotopia in geometria algebrica, Ann. Math. Pura Appl. 100 (1960), 149-186.
  • [V] A. G. Vitushkin, On polynomial transformation of Cn manifolds (Tokyo, 1973), Tokyo Univ. Press, Tokyo, 1975, pp. 415-417.
  • [Wa] S. Wang, A jacobian criterion for separability, J. Algebra 65 ( 1980), 453-494.
  • [Wr1] D. Wright, The amalgamated free product structure of GL2 (k[X1,..., Xn]) and the weak Jacobian Theorem for two variables, J. Pure and Appl. Algebra 12 (1978), 235-251.
  • [Wr2] D. Wright, On the Jacobian Conjecture, Illinois J. Math. 25 (1981), 423-440.