ANNALES POLONICI MATHEMATICI 94.1 (2008) On the Lojasiewicz exponent near the fibre of polynomial mappings by Ha Huy Vui and Nguyen Hong Duc (Hanoi) Abstract. 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. 1. Introduction. Let M,N,L be finite-dimensional real vector spaces and let g : X → N and f : X → L be semialgebraic mappings, where X ⊂ M . For a set S ⊂ X , put L ∞ (g| S ) := sup{ν ∈ R : ∃C,R> 0, ∀x ∈ S (‖x‖≥ R ⇒‖g(x)‖≥ C ‖x‖ ν )}. For λ ∈ L, put L ∞,f →λ (g) := sup{L ∞ (g| f -1 (U ) ): U ⊂ L is a neighbourhood of λ}. Motivated by results of [H], [C-K1], [C-K2], [P], [KOS], . . . on bifurcation values at infinity of polynomial functions, the number L ∞,f →λ (g), called the Lojasiewicz exponent at infinity of g near the fibre f -1 (λ), was introduced and studied in [Sk] and [R-S]. The authors of [R-S] proved that: (i) L ∞,f →λ (g) ∈ Q ∪ {±∞}. (ii) There is a semialgebraic stratification L = S 1 ∪···∪ S j such that the function ν : L ∋ λ →L ∞,f →λ (g) is constant on each stratum S i . Our aim in this paper is to study L ∞,f →λ (g) in the case when f and g are polynomials in two real or complex variables. In this very restric- tive setting we can give complete results about the Lojasiewicz exponent at infinity near the fibre in the complex case. In brief, our results are the fol- lowing. Let f (x,y) be a non-constant monic polynomial in x, i.e. f (x,y)= x d + a 1 (y)x d-1 + ··· + a d (y), where a i ∈ C[y] and deg a i ≤ i. Then: 2000 Mathematics Subject Classification : Primary 14R25; Secondary 32A20, 32S05, 14R25. Key words and phrases : Łojasiewicz exponent near the fibre, Puiseux expansion at infinity. [43] c  Instytut Matematyczny PAN, 2008