## 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

### Citations:

Zbl 0115.10203; Zbl 0651.10022; Zbl 0682.14001
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