We consider Keller's functions, namely polynomial functions f:Cn →Cn with det f(x)=1 at all x εCn. 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 hλ, λ > 1, defined by power series in a ball with center at the origin, such that h′λ(0)=I and hλ(λf(x))=λhλ(x). So each hλconjugates 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)=sdg(x), g′(x)n=0, xεCn, sεC the conjugation hλ for λf is an entire function.
|Original language||English (US)|
|Number of pages||11|
|Journal||ZAMP Zeitschrift für angewandte Mathematik und Physik|
|State||Published - Nov 1 1995|
ASJC Scopus subject areas
- Physics and Astronomy(all)
- Applied Mathematics