Uniform Approximation of Abhyankar Valuation Ideals in Smooth Function Fields

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In this paper we use the theory of multiplier ideals to show that the valuation ideals of a rank one Abhyankar valuation centered at a smooth point of a complex algebraic variety are approximated, in a quite strong sense, by sequences of powers of fixed ideals. Fix a rank one valuation v centered at a smooth point x on an algebraic variety over a field of characteristic zero. Assume that v is Abhyankar, that is, that its rational rank plus its transcendence degree equal the dimension of the variety. Let aₘ denote the ideal of elements in the local ring of x whose valuations are at least m. Our main theorem is that there exists e>0 such that amn is contained in (aₘ₋ₑ)ⁿ for all m and n. This can be viewed as a greatly strengthened form of Izumi's Theorem for Abhyankar valuations centered on smooth complex varieties.