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)].


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