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

*(English)*Zbl 0539.13012Let \(f_ 1,...,f_ n\) 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.

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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\textit{H. Bass} et al., Bull. Am. Math. Soc., New Ser. 7, 287--330 (1982; Zbl 0539.13012)

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