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1
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The Łojasiewicz exponent of subanalytic sets

100%
Spodzieja S.
Annales Polonici Mathematici
|
2005
|
tom 87
|
nr 1
247-263
EN
We prove that the infimum of the regular separation exponents of two subanalytic sets at a point is a rational number, and it is also a regular separation exponent of these sets. Moreover, we consider the problem of attainment of this exponent on analytic curves.
2
Content available remote

The field of Nash functions and factorization of polynomials

100%
Spodzieja S.
Annales Polonici Mathematici
|
1996-1997
|
tom 65
|
nr 1
81-94
EN
The algebraically closed field of Nash functions is introduced. It is shown that this field is an algebraic closure of the field of rational functions in several variables. We give conditions for the irreducibility of polynomials with Nash coefficients, a description of factors of a polynomial over the field of Nash functions and a theorem on continuity of factors.
3
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Some criteria for the injectivity of holomorphic mappings

100%
Spodzieja S.
Annales Polonici Mathematici
|
1991
|
tom 55
|
nr 1
321-323
EN
We prove some criteria for the injectivity of holomorphic mappings.
4
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Errata to "Multiplicity and the Łojasiewicz exponent" (Ann. Polon. Math. 73 (2000), 257-267)

100%
Spodzieja S.
Annales Polonici Mathematici
|
2015
|
tom 115
|
nr 1
99-100
5
Content available remote

Łojasiewicz Exponent of Overdetermined Mappings

64%
Spodzieja S., Szlachcińska A.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
|
tom 61
|
nr 1
27-34
EN
A mapping $F:ℝ ⁿ → ℝ^m$ 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 $F:ℝ ⁿ → ℝ^m$ can be reduced to the case m = n.
6
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The Łojasiewicz gradient inequality in a neighbourhood of the fibre

64%
Gwoździewicz J., Spodzieja S.
Annales Polonici Mathematici
|
2005
|
tom 87
|
nr 1
151-163
EN
Some estimates of the Łojasiewicz gradient exponent at infinity near any fibre of a polynomial in two variables are given. An important point in the proofs is a new Charzyński-Kozłowski-Smale estimate of critical values of a polynomial in one variable.
7
Content available remote

Sum of squares and the Łojasiewicz exponent at infinity

51%
Kurdyka K., Osińska-Ulrych B., Skalski G., Spodzieja S.
Annales Polonici Mathematici
|
2014
|
tom 112
|
nr 3
223-237
EN
Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h₁(x) = ⋯ = h_{r}(x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then $f|_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h(x) = ∑_{i=1}^{r} h²_{i}(x)σ_{i}(x)$, where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) > 0 for x ∈ ℝⁿ. We give an estimate for p in terms of: the degree of f, the degrees of $h_{i}$ and the Łojasiewicz exponent at infinity of $f|_{V}$. We prove a version of the above result for polynomials positive on semialgebraic sets. We also obtain a nonnegative extension of some odd power of f which is nonnegative on an irreducible algebraic set.
8
Content available remote

Salomon's Theorem for polynomials with several parameters

51%
Frontczak M., Skibiński P., Spodzieja S.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 1
107-114
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