Uniform approximation of Abhyankar valuation ideals in smooth function fields

Fix a rank one valuation nu centered at a smooth point x on, an algebraic variety over a field of characteristic zero. Assume that nu 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 k > 0 such that a(mn) is contained in (a(m-k))(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. The proof uses the theory of asymptotic multiplier ideals.