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Resultant and the Łojasiewicz exponent. (English) Zbl 0833.14003
Let $$H=(f,g) : U \to \mathbb{C}^2$$, $$\;0 \in U \subset \mathbb{C}^2$$ a holomorphic mapping having an isolated zero at the origin. The Łojasiewicz exponent of $$H$$ at 0 is the number ${\mathfrak L}_0 (H)=\inf \biggl \{ \nu \in \mathbb{R} : \exists A,B > 0, \forall |z |< B,A |z |^\nu \leq \bigl |H(z) \bigr |\biggr\}$ where $$|z |=\max (|x |, |y |)$$, for $$z=(x,y) \in \mathbb{C}^2$$. – In this paper the authors give an effective formula for the Łojasiewicz exponent for a polynomial mapping $$H=(f,g)$$ in terms of the resultant under the following hypotheses:
(i) $$H^{-1} (0)$$ is a finite fibre;
(ii) $$H(0,y)=0$$ if and only if $$y=0$$;
(iii) $$\deg_y f=\deg f(0,y)$$ or $$\deg_y g=\deg g(0,y)$$;
(iv) $$\text{ord}_y f=\text{ord} f (0,y)$$ or $$\text{ord}_y g=\text{ord} g (0,y)$$.
These hypotheses are not restrictive since (ii), (iii) and (iv) can be obtained by using a linear automorphism, the Łojasiewicz exponent being invariant. In the rest of the paper some examples and extensions of this result are given.

##### MSC:
 14B05 Singularities in algebraic geometry 32S05 Local complex singularities
##### Keywords:
Łojasiewicz exponent; polynomial mapping; resultant
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