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The Jacobian conjecture: Reduction of degree and formal expansion of the inverse. (English) Zbl 0539.13012
Let \(f_ 1,...,f_ n\) be n complex polynomials in n indeterminates \(X_ 1,...,X_ n\) such that the Jacobian determinant \(\det(\partial f_ i/\partial x_ j)\) is a non-zero constant. Then the Jacobian conjecture asserts that the \(X_ j's\) are polynomials in the \(f_ i's\) with complex coefficients. Sometimes this problem is attributed to O. H. Keller who apparently was the first to formulate it [cf. O. H. Keller, Monatsh. Math. Phys. 47, 299-306 (1939; Zbl 0021.15303)].
The paper under review consists of three parts. In the first one, the authors describe the state of this problem (in 1981). They give generalizations, (re)prove partial results and theorems in the folklore of the problem. Moreover, the authors tell the history of the conjecture including a thorough review of the published faulty proofs. This part - which is in fact a good reference for the Jacobian conjecture - is closed with a short discussion of power series methods. - In the second part, the authors reduce the problem to the case \(f_ i=X_ i+H_ i (i=1,...,n)\), where the \(H_ i's\) are homogeneous of degree 3; further, the conjecture is affirmated in some special cases. The same reduction was independently found by A. V. Yagzhev [cf. Sib. Math. J. 21, 747-754 (1981); translation from Sib. Mat. Zh. 21, No 5, 141-150 (1980; Zbl 0466.13009)]. More recently, it was ameliorated: E. Connell and L. van den Dries [cf. J. Pure Appl. Algebra 28, 235-239 (1983; Zbl 0513.13007)] showed that the \(H_ i's\) can be assumed to have integer coefficients and L. M. Drużkowski [cf. Math. Ann. 264, 303-313 (1983; Zbl 0504.13006)] proved that one can even assume that the \(H_ i's\) are powers of linear forms. – In the last part, the authors give an expansion for the formal inverse of the \(f_ i's\), indexed by certain trees, and thus relating the conjecture to combinatorics. Various calculations support the authors’ opinion that the problem could be seriously attacked with these methods. Finally, the detailed bibliography should be mentioned, which apparently contains all relevant papers up to 1981.
Reviewer: G.Angermüller

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13B25 Polynomials over commutative rings
14E05 Rational and birational maps
13B10 Morphisms of commutative rings
14E22 Ramification problems in algebraic geometry
14E07 Birational automorphisms, Cremona group and generalizations
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[1] Allen Altman and Steven Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. · Zbl 0215.37201
[2] S. S. Abhyankar, Lectures on expansion techniques in algebraic geometry, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57, Tata Institute of Fundamental Research, Bombay, 1977. Notes by Balwant Singh. · Zbl 0818.14001
[3] S. S. Abhyankar, Lectures in algebraic geometry, Notes by Chris Christensen, Purdue Univ., 1974.
[4] Shreeram S. Abhyankar, Historical ramblings in algebraic geometry and related algebra, Amer. Math. Monthly 83 (1976), no. 6, 409 – 448. · Zbl 0339.14001
[5] Shreeram S. Abhyankar, William Heinzer, and Paul Eakin, On the uniqueness of the coefficient ring in a polynomial ring, J. Algebra 23 (1972), 310 – 342. · Zbl 0255.13008
[6] Shreeram S. Abhyankar and Tzuong Tsieng Moh, Embeddings of the line in the plane, J. Reine Angew. Math. 276 (1975), 148 – 166. · Zbl 0332.14004
[7] Hyman Bass, Algebraic \?-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. · Zbl 0174.30302
[8] Salomon Bochner and William Ted Martin, Several Complex Variables, Princeton Mathematical Series, vol. 10, Princeton University Press, Princeton, N. J., 1948. · Zbl 0041.05205
[9] L. Andrew Campbell, A condition for a polynomial map to be invertible, Math. Ann. 205 (1973), 243 – 248. · Zbl 0245.13005
[10] Ignacio Canals and Emilio Lluis, On a result of Segre, An. Inst. Mat. Univ. Nac. Autónoma México 10 (1970), 1 – 15 (Spanish). · Zbl 0234.14011
[11] E. H. Connell, A \?-theory for the category of projective algebras, J. Pure Appl. Algebra 5 (1974), 281 – 292. · Zbl 0323.18010
[12] Wolfgang Engel, Ein Satz über ganze Cremona-Transformationen der Ebene, Math. Ann. 130 (1955), 11 – 19 (German). · Zbl 0065.02603
[13] B. L. Fridman, A certain characteristization of the polynomial endomorphisms of \?², Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 319 – 328 (Russian).
[14] Takao Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, 106 – 110. · Zbl 0444.14026
[15] I. J. Good, Generalizations to several variables of Lagrange’s expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc. 56 (1960), 367 – 380. · Zbl 0135.18802
[16] Wolfgang Gröbner, Sopra un teorema di B. Segre, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 31 (1961), 118 – 122 (Italian). · Zbl 0192.27004
[17] C. G. J. Jacobi, De resolutione aequationum per series infinitas, J. Reine Angew. Math. 6 (1830), 257-286. · ERAM 006.0240cj
[18] A. V. Jagžev, On a problem of O.-H. Keller, Sibirsk. Mat. Zh. 21 (1980), no. 5, 141 – 150, 191 (Russian).
[19] S. A. Joni, Lagrange inversion in higher dimensions and umbral operators, Linear and Multilinear Algebra 6 (1978/79), no. 2, 111 – 122. · Zbl 0395.05005
[20] Heinrich W. E. Jung, Über ganze birationale Transformationen der Ebene, J. Reine Angew. Math. 184 (1942), 161 – 174 (German). · Zbl 0027.08503
[21] Ott-Heinrich Keller, Ganze Cremona-Transformationen, Monatsh. Math. Phys. 47 (1939), no. 1, 299 – 306 (German). · Zbl 0021.15303
[22] Arne Magnus, On polynomial solutions of a differential equation, Math. Scand. 3 (1955), 255 – 260 (1956). · Zbl 0067.31605
[23] L. G. Makar-Limanov, The automorphisms of the free algebra with two generators, Funkcional. Anal. i Priložen. 4 (1970), no. 3, 107 – 108 (Russian). · Zbl 0218.13006
[24] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. · Zbl 0441.13001
[25] Gary H. Meisters, Jacobian problems in differential equations and algebraic geometry, Rocky Mountain J. Math. 12 (1982), no. 4, 679 – 705. · Zbl 0523.34050
[26] T. T. Moh, On the Jacobian conjecture and the configurations of roots, J. Reine Angew. Math. 340 (1983), 140 – 212. · Zbl 0525.13011
[27] J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Institute of Fundamental Research, Bombay, 1967. Notes by S. Anantharaman; Tata Institute of Fundamental Research Lectures on Mathematics, No 40. · Zbl 0198.26202
[28] Yoshikazu Nakai and Kiyoshi Baba, A generalization of Magnus’ theorem, Osaka J. Math. 14 (1977), no. 2, 403 – 409. · Zbl 0373.12010
[29] G. C. Wraith, The Jacobian problem, Ganit 1 (1981), no. 1, 1 – 6. · Zbl 0583.13008
[30] Pekka Nousiainen, On the degrees of smooth maps of affine space, Pennsylvania State Univ., preprint, 1981.
[31] Pekka Nousiainen and Moss E. Sweedler, Automorphisms of polynomial and power series rings, J. Pure Appl. Algebra 29 (1983), no. 1, 93 – 97. · Zbl 0545.13003
[32] Susumu Oda, The Jacobian problem and the simply-connectedness of A, Osaka Univ., preprint, 1980.
[33] Michael Razar, Polynomial maps with constant Jacobian, Israel J. Math. 32 (1979), no. 2-3, 97 – 106. · Zbl 0406.13006
[34] I. R. Shafarevich, On some infinite-dimensional groups, Rend. Mat. e Appl. (5) 25 (1966), no. 1-2, 208 – 212.
[35] Beniamino Segre, Corrispondenze di Möbius e trasformazioni cremoniane intere, Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 91 (1956/1957), 3 – 19 (Italian). · Zbl 0079.36801
[36] B. Segre, Forme differenziali e loro integrali, vol. II, Docet, Roma, 1956. · Zbl 0073.07803
[37] Beniamino Segre, Variazione continua ed omotopia in geometria algebrica, Ann. Mat. Pura Appl. (4) 50 (1960), 149 – 186 (Italian). · Zbl 0099.16401
[38] A. G. Vitushkin, On polynomial transformations of \?\(^{n}\), Manifolds — Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 415 – 417.
[39] Stuart Sui Sheng Wang, A Jacobian criterion for separability, J. Algebra 65 (1980), no. 2, 453 – 494. · Zbl 0471.13005
[40] David Wright, The amalgamated free product structure of \?\?\(_{2}\)(\?[\?\(_{1}\),\cdots,\?_{\?}]) and the weak Jacobian theorem for two variables, J. Pure Appl. Algebra 12 (1978), no. 3, 235 – 251. · Zbl 0387.20039
[41] David Wright, On the Jacobian conjecture, Illinois J. Math. 25 (1981), no. 3, 423 – 440. · Zbl 0463.13002
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