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1

Injectivity onto a star-shaped set for local homeomorphisms in n-space

100%

Gorni G., Zampieri G.

Annales Polonici Mathematici

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1994

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tom 59

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nr 2

171-196

EN

We provide a number of either necessary and sufficient or only sufficient conditions on a local homeomorphism defined on an open, connected subset of the n-space to be actually a homeomorphism onto a star-shaped set. The unifying idea is the existence of "auxiliary" scalar functions that enjoy special behaviours along the paths that result from lifting the half-lines that radiate from a point in the codomain space. In our main result this special behaviour is monotonicity, and the auxiliary function can be seen as a Lyapunov function for a suitable dynamical system having the lifted paths as trajectories.

2

On the span invariant for cubic similarity

100%

Gorni G., Tutaj-Gasińska H.

Annales Polonici Mathematici

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2001

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tom 76

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nr 1-2

113-119

EN

Given a real n×n matrix A, we make some conjectures and prove partial results about the range of the function that maps the n-tuple x into the entrywise kth power of the n-tuple Ax. This is of interest in the study of the Jacobian Conjecture.

3

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

100%

Gorni G., Zampieri G.

Annales Polonici Mathematici

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1996

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tom 64

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nr 3

285-290

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.

Ograniczanie wyników

3 Annales Polonici Mathematici

3 Gorni G.

2 Zampieri G.

1 Tutaj-Gasińska H.

1 2001

1 1996

1 1994

Injectivity onto a star-shaped set for local homeomorphisms in n-space

On the span invariant for cubic similarity

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