Abstract
An effective estimate for the local multiplicity of a complete intersection of complex algebraic and Pfaffian varieties is given, based on a local complex analog of the Rolle-Khovanskii theorem. The estimate is valid also for the properly defined multiplicity of a non-isolated intersection. It implies, in particular, effective estimates for the exponents of the polar curves, and the exponents in the Łojasiewicz inequalities for Pfaffian functions. For the intersections defined by sparse polynomials, the multiplicities outside the coordinate hyperplanes can be estimated in terms of the number of non-zero monomials, independent of degrees of the monomials.
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Gabrielov, A. Multiplicities of Pfaffian intersections, and the Łojasiewicz inequality. Selecta Mathematica, New Series 1, 113–127 (1995). https://doi.org/10.1007/BF01614074
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DOI: https://doi.org/10.1007/BF01614074