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

Abstract

Summary.We give a description of the set of points for which the Fedoryuk condition fails in terms of the Łojasiewicz exponent at infinity near a fibre of a polynomial.

Effective formulas for the Łojasiewicz exponent at infinity

“…if P k = P Q , where P ∈ C[y, t], Q ∈ C[y], then P(0, z k ) ≡ 0. This is not true in general (see [24,34,31,29]). To obtain formulas for L ∞ (F ) in the general case we use the Chądzyński-Kollar inequality (1) and reduce the problem to the case of proper and overdetermined polynomial mappings C n…”

“…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]). …”