The Łojasiewicz exponent via the valuative Hamburger-Noether process

Abstract

Abstract. Let k be an algebraically closed field of any characteristic. We apply the Hamburger-Noether process of successive quadratic transformations to show the equivalence of two definitions of the Łojasiewicz exponent L(a) of an ideal a ⊂ k [[x, y]].

A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularities

“…This is also an easy consequence of Corollary 7 below, so we prove it in Corollary 5. Still more generally, using Theorem 6 we can show (see [3] for a proof in dimension 2): Proposition 4. Let K be an algebraically closed field and K…”

“…Of crucial importance is the following parametric version of the well-known Valuative Criterion of Integral Dependence (see [9] or Corollary 7 for the complex analytic setting; an alternative proof of the theorem stated below, valid in dimension 2 and based on so-called Hamburger-Noether process, can be found in [3,Theorem 21]): Theorem 6. Let K be a field and I be an ideal in the ring K[[z]] = K[[z 1 , .…”