The gradient of a polynomial at infinity

Abstract

We give a description of growth at infinity of the gradient of a polynomial in two complex variables near any of its fiber.

On the Łojasiewicz Exponent near the Fibre of a Polynomial

“…where Φ is a meromorphic mapping defined in a neighbourhood of ∞ in C, deg Φ > 0 and deg(f − λ) • Φ < 0. They prove the equivalence of definitions (1) and (2) in case n = 2. In this paper we show that definitions (1) and (2) are equivalent for any n ≥ 2 and λ ∈ C (Theorem 2.1 in Section 2).…”

Łojasiewicz exponent near the fibre of a mapping

“…This notion was introduced by Ha [7] in the case of complex polynomial functions in two variables (see also [3], [5]). …”

“…Chadzyński and Krasiński ( [3], Corollary 4.7) proved that for a complex polynomial f in two variables with deg f > 0 there exists c f ∈ Q with c f 0 such that…”

Effective formulas for the Łojasiewicz exponent at infinity

“…For m = n some results are known. If n = m = 2, effective formulas for L ∞ (F ) in terms of the discriminant of a polynomial were given by J. Chądzyński and T. Krasiński (see [32,19,20,23]). Płoski [22,Corollary 2.6] gave a formula for L ∞ (F ) in terms of characteristic polynomials, provided n = m and F is proper.…”

“…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]). …”