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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
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