On the Łojasiewicz exponent near the fibre of polynomial mappings
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
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 such that r is the Łojasiewicz exponent of grad(f) near the fiber . 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...
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