### Abstract

We consider Keller's functions, namely polynomial functions f:C^{n} →C^{n} with det f(x)=1 at all x εC^{n}. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open. Without loss of generality assume f(0)=0 and f'(0)=I. We study the existence of certain mappings h_{λ}, λ > 1, defined by power series in a ball with center at the origin, such that h′_{λ}(0)=I and h_{λ}(λf(x))=λh_{λ}(x). So each h_{λ}conjugates λf to its linear part λI in a ball where it is injective. We conjecture that for Keller's functions f of the homogeneous form f(x)=x +g(x), g(sx)=s^{d}g(x), g′(x)^{n}=0, xεC^{n}, sεC the conjugation h_{λ} for λf is an entire function.

Original language | English (US) |
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Pages (from-to) | 872-882 |

Number of pages | 11 |

Journal | ZAMP Zeitschrift für angewandte Mathematik und Physik |

Volume | 46 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1995 |

### ASJC Scopus subject areas

- Mathematics(all)
- Physics and Astronomy(all)
- Applied Mathematics

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## Cite this

*ZAMP Zeitschrift für angewandte Mathematik und Physik*,

*46*(6), 872-882. https://doi.org/10.1007/BF00917874