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A global Łojasiewicz inequality for algebraic varieties. (English) Zbl 0762.14001

For \(X\), the zero locus of polynomials \(f_1,\ldots,f_k\) in \(n\) complex variables, the Łojasiewicz inequality [see S. Łojasiewicz, Studia Math. 18, 87–136 (1959; Zbl 0115.10203)] gives an upper bound for the distance to \(X\). In [J. Am. Math. Soc. 1, No. 2, 311–322 (1988; Zbl 0651.10022)], W. D. Brownawell has proved that the exponent in this inequality is bounded by \(d^{\min(n,k)}\), where \(d=\max(3,\deg f_i)\). The aim of this paper is to find the best possible exponent in terms of \(\deg f_i\). In particular the authors show results for any algebraically closed field and give also improvements of Brownawell’s results. The proofs are based on Brownawell’s version of the effective Nullstellensatz [see J. Kollár, J. Am. Math. Soc. 1, No. 4, 963–975 (1988; Zbl 0682.14001)].

MSC:

14A05 Relevant commutative algebra
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
14Q20 Effectivity, complexity and computational aspects of algebraic geometry
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