We give an effective formula for the local Łojasiewicz exponent of a polynomial mapping. Moreover, we give an algorithm for computing the local dimension of an algebraic variety.Keywords Łojasiewicz exponent · Effective formulas · Germ of algebraic set · Dimension. For the properties of the mappings finite at 0 see [19].The local Łojasiewicz exponent of a mapping F : (C n , 0) → (C m , 0) is defined to be the infimum of the set of all exponents ν ∈ R in the Łojasiewicz inequality |F(z)| ≥ C|z| ν as |z| → 0 for some constant C > 0, and denoted by L 0 (F).The local Łojasiewicz exponent is an important tool in singularity theory (see [3][4][5]11,[13][14][15][16][17][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][40][41][42]). For instance, this exponent is strongly related to the order function ν I ( f ) (see [28]) and polar quotients (see [16,17,32,33,42]). The degree of C 0 -sufficiency of the jet determined by a holomorphic function f : (C n , 0) → (C, 0) is equal to [L 0 (grad f )] + 1

Abstract. Let g : X → R k and f : X → R m , where X ⊂ R n , be continuous semi-algebraic mappings, and λ ∈ R m . We describe the optimal exponent θ =: L ∞,f →λ (g) for which the Lojasiewicz inequality |g(x)| C|x| θ holds with C > 0 as |x| → ∞ and f (x) → λ. We prove that there exists a semi-algebraic stratificationis constant on each stratum S i . We apply this result to describe the set of generalized critical values of f .

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a b s t r a c tWe give effective formulas for the Łojasiewicz exponent at infinity of an arbitrary complex polynomial mapping.

We give, in terms of the Łojasiewicz inequality, a sufficient condition for C k -mappings germs of non-isolated singularity at zero to be isotopical.