A mountain pass to the Jacobian Conjecture

Canadian Mathematical Bulletin

Volumn 41, Issue 4, 1998, Pages 442-451

  Chamberland, Marc ^a   Meisters, Gary ^b  

^a MCMASTER UNIVERSITY   (Canada)

^b UNIVERSITY OF NEBRASKA LINCOLN   (United States)

Author keywords

C1 maps; Injectivity; Jacobian Conjecture; Mountain Pass; Polynomial maps

Indexed keywords

EID: 0000763169     PISSN: 00084395     EISSN: None     Source Type: Journal    
DOI: 10.4153/CMB-1998-058-4     Document Type: Article

References (33)

1. A. Ambrosetti and P. Rabinowitz, Dual Variational Methods in Critical Point Theory and Applications. J. Funct. Anal. 14(1973), 349-381.
2. H. Bass, E. H. Connell and D. Wright, The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse. Bull. Amer. Math. Soc. 7(1982), 287-330.
3. A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties. Proc. Amer. Math. Soc. 13(1962), 200-203.
4. H. Brezis and L. Nirenberg, Remarks on Finding Critical Points. Comm. Pure Appl. Math. 44(1991), 939-963.
5. L. A. Campbell, Rational Samuelson maps are univalent. J. Pure Appl. Algebra 92(1994), 227-240.
6. L. A. Campbell, Partial Properness and Real Planar Maps. Appl. Math. Lett. (5) 9(1996), 99-105.
7. 3. Univ. of Nijmegen, The Netherlands, Mathematics Dept. Report 9551 (Nov. 1995). The final blow to the Markus-Yamabe Conjecture in dimensions n > 2; to appear in Adv. Math.
8. A. van den Essen via B. A. Coomes, Details of Pinchuk's degree (10,25) counterexample to the Strong Real Jacobian Conjecture. Personal communication from Essen via Coomes, June 3, 1994.
9. EX preprint, Spring 1995. Characterizes analytic maps which are globally analytically conjugate to their linear part at a fixed point. See also Rosay and Rudin [29].
10. A. van den Essen, Counterexample to a conjecture of Meisters. Nijmegen Report 9441 (Oct. 1994). The Conjugation Conjecture for F = X - H is false.
11. A. van den Essen and E. Hubbers, Chaotic Polyomorphisms: Counterexamples to Several Conjectures. Nijmegen Report 9549 (November 1995). Adv. in Appl. Math. (3) 18(1997), 382-388.
12. A. van den Essen (ed.), Automorphisms of Affine Spaces. Proceedings of a Conference on Polynomial Automorphisms, at the Princess Beach Resort Casino, Willemstad, Curaçao, The Netherlands Antilles, July 4-8, 1994. Kluwer Academic Publishers, 1995.
13. D. Gale and H. Nikaidô, The Jacobian matrix and global univalence. Math. Ann. 159(1965), 81-93.
14. W. B. Gordon, On the Diffeomorphisms of Euclidean Space. Amer. Math. Monthly 79(1972), 755-759. (See also Addendum (6) 80(1973), 674-675; and Application (1) 84(1977), 28-29.)
15. G. Gorni and G. Zampieri, On cubic-linear polynomial mappings. UDMI/43/96/RR (September 1996), 21 pages, Udine, Italy; Nederl. Akad. Wetensch. Indag. Math., to appear.
16. O.-H. Keller, Ganze Cremona Transformationen. Monatsh Math. 47(1939), 299-306, Items 6 and 7 in the table on page 301.
17. G. H. Meisters, Jacobian problems in differential equations and algebraic geometry. Rocky Mountain J. Math. 12(1982), 679-705.
18. _, Inverting polynomial maps of n-space by solving differential equations. In: Delay and Differential Equations, Proceedings in Honor of George Seifert, Ames, Iowa, October 18-19, 1991 (Eds.: Fink/Miller/Kliemann). World Scientific, Singapore-River Edge, NJ-London-Hong Kong, 1992, 107-166.
19. _, Wanted: A Bad Matrix. Amer. Math. Monthly 102(1995), 546-550
20. _, PolyMapBib 1997, 30 pp. World Wide Web, http://www.math.unl.edu/̃gmeister/.
21. _, A Biography of the Markus-Yamabe Conjecture. In: Aspects of Mathematics: Algebra, Geometry and Several Complex Variables. Proc. of Conf. June 10-13, 1996, at HKU, to appear.
22. Noble and Daniel, Applied Linear Algebra. 2nd edn, Prentice-Hall, 1977, 431, Theorem 11.7.
23. R. S. Palais and S. Smale, A generalized Morse Theory. Bull. Amer. Math. Soc. 70(1964), 165-171.
24. S. I. Pinchuk, Counterexample to the Strong Real Jacobian Conjecture. Math. Zeit. 217(1994), 1-4.
25. R. Plastock, Homeomorphisms between Banach spaces. Trans. Amer. Math. Soc. 200(1974), 169-183.
26. P. Pucci and J. Serrin, The Structure of the Critical Set in the Mountain Pass Theorem. Trans. Amer. Math. Soc. 299(1987), 115-132.
27. P. J. Rabier, On Global Diffeomorphisms of Euclidean Space. Nonlinear Anal. 21(1993), 925-947.
28. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986.
29. n. Trans. Amer. Math. Soc. 310(1988), 47-86. Compare Theorem in the Appendix with Bo Deng [9].
30. W. Rudin, Injective Polynomial Maps are Automorphisms. Amer. Math. Monthly 102(1995), 540-543.
31. B. Smyth and F. Xavier, Injectivity of Local Diffeomorphisms from Nearly Spectral Conditions. J. Differential Equations 130(1996), 406-414.
32. D. Wright, On the Jacobian Conjecture. Illinois J. Math.25(1981), 423-440.
33. A. V. Yagzhev, Keller's Problem. Siberian Math. J. 21(1981), 747-754.