Uniform approximation of Abhyankar valuation ideals in smooth function fields

Fix a rank one valuation ν centered at a smooth point x on an algebraic variety over a field of characteristic zero. Assume that ν is Abhyankar, that is, that its rational rank plus its transcendence degree equal the dimension of the variety. Let a_m 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.