On the Łojasiewicz Exponent near the Fibre of a Polynomial

Abstract

The equivalence of the definitions of the Łojasiewicz exponent introduced by Ha and by Chądzyński and Krasiński is proved. Moreover we show that if the above exponents are less than −1 then they are attained at a curve meromorphic at infinity.

Effective formulas for the Łojasiewicz exponent at infinity

“…The exponent L ∞ (F ) is intimately related to the properties of properness (a mapping is proper if the inverse image of any compact set is compact) and injectivity of polynomial mappings (see [17,3,[18][19][20][21][22]). The Łojasiewicz exponent of the gradient is also related to the triviality of polynomials at infinity (see [23][24][25][26][27][28][29][30][31]). …”

The Fedoryuk Condition and the Łojasiewicz Exponent near a Fibre of a Polynomial

“…Chądzyński and Krasiński [2] for n = 2 and Skalski [8] for arbitrary n proved that this definition is equivalent to the following definition introduced by Ha [4]:…”