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6 Bulletin of the Polish Academy of Sciences. Mathematics

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Multiplicity and Semicontinuity of the Łojasiewicz Exponent

Rodak T., Różycki A., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

2016

Vol. 64, no. 1

55--62

We give an effective formula for the improper isolated multiplicity of a polynomial mapping. Using this formula we construct, for a given deformation of a holomorphic mapping with an isolated zero at zero, a stratification of the space of parameters such that the Łojasiewicz exponent is constant on each stratum.

Łojasiewicz Exponent of Overdetermined Mappings

Spodzieja S., Szlachcińska A.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 1

27--34

A mapping F:Rn→Rm is called overdetermined if m>n. We prove that the calculations of both the local and global Łojasiewicz exponent of a real overdetermined polynomial mapping F:Rn→Rm can be reduced to the case m=n.

The Fedoryuk condition and the Łojasiewicz exponent near a fibre of a polynomial

Rodak T.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 3

227-229

We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.

On the Łojasiewicz exponent near the fibre of a polynomial

Skalski G.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 3

231-236

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than -1 then they are attained at a curve meromorphic at infinity.

On some characterization of proper polynomial mappings

Rodak T., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 2

157-164

It is well known that a proper, in the classical topology, polynomial mapping is closed in the Zariski topology. In the paper we prove that the inverse is true. Namely, any non-constant polynomial mapping from [C^n] into [C^m] which is closed in the Zariski topology is proper in the classical topology.

On the Łojasiewicz exponent of the gradient of holomorphic functions

Bogusławska M.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 4

337-343

We show that the Łojasiewicz exponent L[sub o] (grad f) at zero of the gradient of a distinguished pseudo-polynomial f [...] is attained on the zero-set of f'[sub y].