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1996 | 64 | 3 | 285-290
Tytuł artykułu

Yagzhev polynomial mappings: on the structure of the Taylor expansion of their local inverse

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EN
Abstrakty
EN
It is well known that the Jacobian conjecture follows if it is proved for the special polynomial mappings $f:ℂ^n → ℂ^n$ of the Yagzhev type: f(x) = x - G(x,x,x), where G is a trilinear form and $det f'(x) ≡ 1. Drużkowski and Rusek [7] showed that if we take the local inverse of f at the origin and expand it into a Taylor series $∑_{k≥1}Φ_k$ of homogeneous terms $Φ_k$ of degree k, we find that $Φ_{2m+1}$ is a linear combination of certain m-fold "nested compositions" of G with itself. If the Jacobian Conjecture were true, $f^{-1}$ should be a polynomial mapping of degree $≤ 3^{n-1}$ and the terms $Φ_k$ ought to vanish identically for $k > 3^{n-1}$. We may wonder whether the reason why $Φ_{2m+1}$ vanishes is that each of the nested compositions is somehow zero for large m. In this note we show that this is not at all the case, using a polynomial mapping found by van den Essen for other purposes.
Słowa kluczowe
Rocznik
Tom
64
Numer
3
Strony
285-290
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-10-23
poprawiono
1996-02-07
Twórcy
  • Università di Udine, Dipartimento di Matematica e Informatica, Via delle Scienze 208, 33100 Udine, Italy
  • Università di Messina, Dipartimento di Matematica, Salita Sperone 31, 98166 Sant'Agata, Messina, Italy
Bibliografia
  • [1] H. Bass, E. Connell and D. Wright, The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982), 287-330.
  • [2] A. Białynicki-Birula and M. Rosenlicht, Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc. 13 (1962), 200-203.
  • [3] B. Deng, Automorphic conjugation, global attractor, and the Jacobian conjecture, University of Nebraska-Lincoln, 1995.
  • [4] B. Deng, G. H. Meisters and G. Zampieri, Conjugation for polynomial mappings, Z. Angew. Math. Phys. 46 (1995), 872-882.
  • [5] L. M. Drużkowski, An effective approach to Keller's Jacobian conjecture, Math. Ann. 264 (1983), 303-313.
  • [6] L. M. Drużkowski, The Jacobian conjecture, Institute of Mathematics, Polish Academy of Sciences, preprint 492 (1991).
  • [7] L. M. Drużkowski and K. Rusek, The formal inverse and the Jacobian conjecture, Ann. Polon. Math. 46 (1985), 85-90.
  • [8] A. van den Essen (ed.), Automorphisms of Affine Spaces, Proc. of the Curaçao Conference, Kluwer Acad. Publ., 1985.
  • [9] G. Gorni and G. Zampieri, On the existence of global analytic conjugations for polynomial mappings of Yagzhev type, J. Math. Anal. Appl., to appear.
  • [10] O. H. Keller, Ganze Cremona Transformationen, Monatsh. Math. Phys. 47 (1939), 299-306.
  • [11] W. Rudin, Injective polynomial maps are automorphisms, Amer. Math. Monthly 102 (1995), 540-543.
  • [12] A. V. Yagzhev, Keller's problem, Siberian Math. J. 21 (1980), 747-754.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-apmv64z3p285bwm
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