## Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings.(English)Zbl 0942.32003

The main results of the paper are:
A. Let $$X$$ and $$Y$$ be arbitrary algebraic subsets of $${\mathbb {CP}}^n$$ endowed with a distance function $$\rho$$ induced by a Riemannian metric. If $$X\cap Y\neq\emptyset$$ then there exists a constant $$c>0$$ such that for $$z\in{\mathbb {CP}}^n$$ $\rho(z,X)+\rho(z,Y)\geq c\rho(z,X\cap Y)^{{\deg X}\cdot{\deg Y}}.$ Here $$X$$ and $$Y$$ can have different irreducible components the degree being defined as the sum of the degrees of the irreducible components. For $$X$$, $$Y$$ hypersurfaces in $${\mathbb C}^n$$ separation results of this type where obtained by S. Ji, J. Kollár and B. Shiffman [Trans. Am. Math. Soc. 329, No. 2, 813-818 (1992; Zbl 0762.14001)].
B. Let $$F=(F_1,\ldots,F_m):{\mathbb C}^n\rightarrow{\mathbb C}^m$$ such that $$F^{-1}(0)$$ is finite. Then the Łojasiewicz exponent $${\mathcal L}_{\infty}(F)$$ of $$F$$ satisfies the inequality: ${\mathcal L}_{\infty}(F)\geq d_m-B(d_1,\ldots,d_m,n)+ \sum_{b\in F^{-1}(0)}\mu_b(F)$ Here $$d_j=\deg F_j$$ and it is assumed that $$d_1\geq\ldots\geq d_m>0$$. Also $$\mu_b(F)$$ is the multiplicity of $$F$$ at the isolated zero $$b$$ and $$B(d_1,\ldots,d_m,n)= d_1 \cdot \ldots \cdot d_{m-1}\cdot d_m$$ if $$m\leq n$$ and equals $$d_1 \cdot \ldots \cdot d_{n-1}\cdot d_m$$ if $$n<m$$.
This inequality is an improvement of a result of J. Kollár [J. Am. Math. Soc. 1, No. 4, 963-975 (1988; Zbl 0682.14001)].
One of the main points in the proof of A is a local separation result obtained by the first author in her thesis [Ann. Pol. Math. 69, No. 3, 287-299 (1998; Zbl 0924.32005)].

### MSC:

 32B20 Semi-analytic sets, subanalytic sets, and generalizations 14P15 Real-analytic and semi-analytic sets

### Citations:

Zbl 0762.14001; Zbl 0682.14001; Zbl 0924.32005
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