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1

Multiplicity and Semicontinuity of the Łojasiewicz Exponent

Rodak T., Różycki A., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 1

55--62

EN

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.

2

Łojasiewicz Exponent of Overdetermined Mappings

Spodzieja S., Szlachcińska A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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Vol. 61, no 1

27--34

EN

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.

3

Polynomial imaginary decompositions for finite extensions of fields of characteristic zero

Nowicki A., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 2

157-168

EN

Let k be a field of characteristic zero, L = k[xi] a finite field extension of k of degree m > 1. If f is a polynomial in one variable over L, then there exist unique polynomials u0,..., um-1 belonging to k[x0,..., xm-l] such that f(x0 + xix1 + ...xi^m-1 xm-l) = uO + xiu1 + ...xi^m-1 um-1. We prove that for u0, ..., um-1is an element of k[xo,..., xm-1) there exists f for which the above holds if and only if u0, ..., um-1satisfy some generalization of the Cauchy-Riemann equations. Moreover, we show that if f is not an element of L, then the polynomials u0, ... ,um-1 are algebraically independent over k and they have no common divisors in k[xo,... ,Xm-1) of positive degree. Some other properties of polynomials u0,..., um-1 are also given.

4

Łojasiewicz inequalities at infinity for the gradient of a polynomial

Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2002

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Vol. 50, no 3

273-281

EN

Some Łojasiewicz inequalities at infinity for the gradient of a polynomial are given.

5

On some characterization of proper polynomial mappings

Rodak T., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

157-164

EN

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.

6

On factorization of polynomials with holomorphic coefficients

Frontczak M., Skibiński A., Spodzieja S.

Bulletin of the Polish Academy of Sciences. Mathematics

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1998

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Vol. 46, no 1

39-54

EN

In the paper some equivalent conditions to pointwise reducibility of polynomials with holomorphic coefficients (in an arbitrary connected set) are given.