Abstract
We consider Keller's functions, namely polynomial functionsf:C n →C n with detf(x)=1 at allx εC n. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.
Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh λ, λ > 1, defined by power series in a ball with center at the origin, such thath′λ(0)=I andh λ(λf(x))=λh λ(x). So eachh λ conjugates λf to its linear part λI in a ball where it is injective.
We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s dg(x),g′(x)n=0,xεC n,sεC the conjugationh λ for λf is anentire function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, Berlin 1983.
H. Bass, E. Connell, and D. Wright,The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc.7, 287–330 (1982).
A. Bialynicki-Birula and M. Rosenlicht,Injective morphisms of real algebraic varieties, Proc. Amer. Math. Soc.13, 200–203 (1962).
G. De Marco, G. Gorni, and G. Zampieri,Global inversion of functions: an introduction, Nonlinear Diff. Equat. and Appl.1, 229–248 (1994).
L. Druźkowski,An effective approach to Keller's Jacobian conjecture, Math. Ann.264, 303–313 (1983).
S. Friedland and J. Milnor,Plane polynomial automorphisms, Ergod. Th. and Dynam. Syst.9, 67–99 (1989).
O. H. Keller,Ganze Cremona trasformationen, Monatshefte für Mathematik und Physik47, 299–306 (1939).
G. H. Meisters,Inverting polynomial maps of n-space by solving differential equations, in Fink, Miller, Kliemann, eds.,Delay and Differential Equations: Proc. in Honor of George Seifert on his retirement, World Sci. Publ. Co., 107–166 (1992).
G. H. Meisters and C. Olech,Global stability, injectivity, and the Jacobian conjecture, de Gruyter, Berlin 1994. Proc. First World Congress of Nonlinear Analysts, Tampa, Florida. (Ed. Lakshmikantham).
D. J. Newman,One-one polynomial maps, Proc. Amer. Math. Soc.11, 867–870 (1960).
K. Rusek,A geometric approach to Keller's Jacobian conjecture, Math. Ann.264, 315–320 (1983).
J. Sotomayor,Inversion of smooth mappings, Z. angew. Math. Phys. ZAMP41, 306–310 (1990).
A. V. Yagzhev,Keller's problem, Siberian Math. J.21, 747–754 (1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Deng, B., Meisters, G.H. & Zampieri, G. Conjugation for polynomial mappings. Z. angew. Math. Phys. 46, 872–882 (1995). https://doi.org/10.1007/BF00917874
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00917874