# Lojasiewicz inequalities, topological equivalences and newton polyhedra

Definition 4.1. Let f; g : Kn ! K be polynomial functions. We say that the family f + tg is analytically (resp., smooth) trivial at infinity along the interval [0; 1] if there exist a neighborhood of infinity Ω0 ⊂ Kn and a continuous mapping Φ: [0; 1] × Ω0 ! Kn; (t; x) 7! Φ(t; x); such that the following conditions are satisfied (a) Φ0(x) = x for x 2 Ω0; (b) for any t 2 [0; 1]; the mapping Φt : Ω0 ! Φt(Ω0) is a real analytic diffeomorphism (resp., C1-diffeomorphism) and lim x!1 Φt(x) = 1; (c) f(Φt(x)) + tg(Φt(x)) = f(x) for x 2 Ω0 and t 2 [0; 1]; where the mapping Φt : Ω0 ! Kn is defined by Φt(x) := Φ(t; x) for x 2 Ω0 and t 2 [0; 1]: With the above definitions, the main result is as follows 73 trang | Chia sẻ: tueminh09 | Ngày: 22/01/2022 | Lượt xem: 118 | Lượt tải: 0
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