A note on the Jacobian condition and two points at infinity

Authors:
James H. McKay and Stuart Sui Sheng Wang

Journal:
Proc. Amer. Math. Soc. **111** (1991), 35-43

MSC:
Primary 14E07; Secondary 13B10, 14E20

MathSciNet review:
1034887

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Abstract: If two polynomials and satisfy the Jacobian condition and the Newton polygon of has an edge of negative slope, then the sum of terms of along this edge has at most two distinct irreducible factors and their exponents must be different. Moreover, the slope is either a (negative) integer and the edge touches the -axis or a (negative) Egyptian fraction and the edge touches the -axis. Furthermore, there is an elementary automorphism which reduces the size of the Newton polygon.

**[1]**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. MR**542446****[2]**Harry Appelgate and Hironori Onishi,*The Jacobian conjecture in two variables*, J. Pure Appl. Algebra**37**(1985), no. 3, 215–227. MR**797863**, 10.1016/0022-4049(85)90099-4**[3]**Z. Charzyński, J. Chadzyński and P. Skibiński,*A contribution to Keller's Jacobian conjecture*, Seminar on deformations (Proceedings, Łódź--Warsaw 1982/84, Lecture Notes in Mathematics 1165), Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1985, pp. 36-51.**[4]**L. G. Makar-Limanov, 1969, unpublished.**[5]**James H. McKay and Stuart Sui Sheng Wang,*An elementary proof of the automorphism theorem for the polynomial ring in two variables*, J. Pure Appl. Algebra**52**(1988), no. 1-2, 91–102. MR**949340**, 10.1016/0022-4049(88)90137-5**[6]**T. T. Moh,*On the Jacobian conjecture and the configurations of roots*, J. Reine Angew. Math.**340**(1983), 140–212. MR**691964****[7]**A. G. Vitushkin,*On polynomial transformations of 𝐶ⁿ*, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 415–417. MR**0369367**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1991-1034887-3

Keywords:
Jacobian condition (Jacobian hypothesis),
Newton polygon,
points at infinity

Article copyright:
© Copyright 1991
American Mathematical Society