Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals

Abstract

Let g : (C n , 0) → (C n , 0) be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I 1 , . . . , I n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions.

Arnold's problem on monotonicity of the Newton number for surface singularities

“…Since W 1 = È(L,R 2 ), W 1 = È(L, R 2 ) are two perpendicular facets of ∆(R 1 , R 2 , R 2 , P), we can continue the above equality = 3! vol 3…”

Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

“…where J denotes the integral closure of a given ideal J of O n . The above expression was one of the motivations that lead the first author to introduce in [5] the notion of Lojasiewicz exponent of a set of ideals (see Definition 2.6). By substituting m by a proper ideal J of O n in (2) we obtain what is known as the Lojasiewicz exponent of I with respect to J (see (10), (11) and [17]).…”

Mixed Łojasiewicz exponents and log canonical thresholds of ideals

“…This fact is one of the motivations of the definition in [4] of the notion of Lojasiewicz exponent of a set of ideals. The main tool used for this definition is the mixed multiplicity of n ideals in a local ring of dimension n.…”

“…where J is the monomial ideal given by J = ⟨x 14 , y 6 , z 4 , y 5 z, xy 6 ⟩. Obviously J ⊆ J(f t ).…”

“…, I n ) in the context of Lojasiewicz exponent. This was the motivation of the first author to introduce the Lojasiewicz exponent of a set of ideals in [4]. If (R, m) denotes a local ring of dimension n and I 1 , .…”