Extensions of regular mappings and the Łojasiewicz exponent at infinity

Osińska, Beata

doi:10.1016/j.bulsci.2010.11.002

Cited by 2 publications

(6 citation statements)

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“…We obtain also a strengthening of the Osińska's result [7,Theorem 1.7] in the complex case (see Remark 4.1 in Section 4).…”

Section: Introductionmentioning

confidence: 48%

“…To make this article independent we give full proof of the theorem, although some parts of the proof can be found in [7]. We do so in order not to miss subtle reasoning, specific consideration of the field of real numbers.…”

Section: Proof Of Theorem 11mentioning

confidence: 97%

“…In the proof of Theorem 1.2 we will need the following proposition (cf. [7,Proposition 4.4] in the complex case).…”

Section: Preliminariesmentioning

confidence: 97%

“…B. Osińska in the article [7,Theorem 1.6] proved that every such a mapping F has a polynomial extension G : C n → C m with the same Łojasiewicz exponent at infinity, i.e. L ∞ (G) = L ∞ (F |V ).…”

Section: Introductionmentioning

confidence: 98%

“…The exponent L ∞ (F ) is an important invariant and tool in the study of behaviour of polynomial mappings at infinity and in optimization (for references, see for instance [7,9,10]). …”

Section: Introductionmentioning

confidence: 99%

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Extensions of real regular mappings and the Łojasiewicz exponent at infinity

Osińska-Ulrych

Skalski

Spodzieja

2013

Bulletin des Sciences Mathématiques

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Let V ⊂ R n be an algebraic set of positive dimension and let L ∞ (F ) be the Łojasiewicz exponent at infinity of a regular mapping F : V → R m . We prove that F has a polynomial extension G : R n → R m such that L ∞ (G) = L ∞ (F ). Moreover, we give an estimate of the degree of this extension. Additionally, we prove that if dim V < n − 2, then for any β ∈ Q, β < L ∞ (F ) the mapping F has a polynomial extension G with L ∞ (G) = β. We also estimate its degree.

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“…We obtain also a strengthening of the Osińska's result [7,Theorem 1.7] in the complex case (see Remark 4.1 in Section 4).…”

Section: Introductionmentioning

confidence: 48%

Section: Proof Of Theorem 11mentioning

confidence: 97%

“…In the proof of Theorem 1.2 we will need the following proposition (cf. [7,Proposition 4.4] in the complex case).…”

Section: Preliminariesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

“…The exponent L ∞ (F ) is an important invariant and tool in the study of behaviour of polynomial mappings at infinity and in optimization (for references, see for instance [7,9,10]). …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations