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A geometrical approach to the Jacobian conjecture for \(n=2\). (English) Zbl 1128.14301
Summary: Let us recall the Jacobian conjecture: “Jacobian Conjecture. Let \(F: \mathbb C^n \to \mathbb C^n\) be a polynomial map. Suppose that, for every point \(x \in \mathbb C^n\), the derivative \(F' (x)\) is invertible. Then the map \(F\) is invertible.”
In this lecture we describe a geometrical approach to this conjecture when \(n=2\).

MSC:
14E07 Birational automorphisms, Cremona group and generalizations
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14E05 Rational and birational maps
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References:
[1] E. Artal, Une demonstration Geometrique du theoreme d’AbhyanJcar-Moh,
[2] S. Abhyankar, T. T. Moh, Embeddings of the line in the plane, Jr. Angw. Reine Math. 27 (1975), 148-166. · Zbl 0332.14004
[3] H. Bass, E. H. Connell, D. Wright, The Jacobian Conjecture: reduction of degree and forma expansion of the inverse, Bull. Amer. Math. Soc. 7, (1982), 287-330. · Zbl 0539.13012
[4] S. A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Inv. Math. 9 (1988), 217-241. · Zbl 0658.32005
[5] Ha Huy Vui: Le Dung Trang, Sur la topologie des polynmes complexes, Acta Math. Vietn. (1984), 21-32. · Zbl 0597.32005
[6] Le Dung Trang, F. Michel, C. Weber, Sur le comportement des polaires associees au germes de courbes planes, Comp. Math. 72 (1989), 87-113. · Zbl 0705.32021
[7] T. T. Moh, On analytic irreducibility at oo of a pencil of curves, Proc. Amer. Math. Soc. 4 (1974), 22-24. · Zbl 0309.14011
[8] D. Mumford, Algebraic Geometry I, Complex Algebraic Varieties, Grundlehren der math. Wis senschaften 221, Springer Verlag (1976). · Zbl 0356.14002
[9] S. Yu. Orevkov, On three-sheeted polynomial mappings of C2, Math. USSR Izvestiya, 2 (1987), 587-596. · Zbl 0633.13005
[10] A. G. Vitushkin, Certain examples in connections with problems about polynomial transfer mations of Cn, Math. USSR Izvestiya, 5 (1971), 278-288. · Zbl 0248.32020
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