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On the Łojasiewicz exponent near the fibre of polynomial mappings

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Vui H., Duc N.

Annales Polonici Mathematici

2008

tom 94

nr 1

43-52

We give the formula expressing the Łojasiewicz exponent near the fibre of polynomial mappings in two variables in terms of the Puiseux expansions at infinity of the fibre.

Łojasiewicz exponent of the gradient near the fiber

100%

Vui H., Duc N.

Annales Polonici Mathematici

2009

tom 96

nr 3

197-207

It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber $f^{-1}(t₀)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}(t₀)$. We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula computing the Łojasiewicz exponent of the gradient near a fiber of a polynomial in two real variables. In particular, this gives, in the case of two real variables, a way to tell whether a given value is an asymptotic critical value or not.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Duc N.

2 Vui H.

1 2009

1 2008

Wyniki wyszukiwania

On the Łojasiewicz exponent near the fibre of polynomial mappings

Łojasiewicz exponent of the gradient near the fiber