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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.

14B05 Singularities in algebraic geometry
32S05 Local complex singularities
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