Jacobian Ideals and the Newton Non-Degeneracy Condition

Abstract

In this paper we extract some conclusions about Newton non-degenerate ideals and the computation of Lojasiewicz exponents relative to this kind of ideal. This motivates us to study the Newton non-degeneracy condition on the Jacobian ideal of a given analytic function germ f : (C n , 0) → (C, 0). In particular, we establish a connection between Newton non-degenerate functions and functions whose Jacobian ideal is Newton non-degenerate.

Mixed Łojasiewicz exponents and log canonical thresholds of ideals

“…Thus the integral closure of J(f 0 ) is a monomial ideal. That is J(f 0 ) = ⟨x 14 , y 7 , xy 6 , z 4 ⟩.…”

“…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 ).…”

“…. , e n denotes the canonical basis in R n (see for instance [6,Corollary 3.6]). It is also known that the log canonical threshold of I, denoted by lct(I), which is another fundamental number associated to ideals of O n (see Section 5), verifies that 1 lct(I) = min{λ > 0 : λ(1, .…”

Mixed Newton numbers and isolated complete intersection singularities

“…Example 6.10. Let J 1 and J 2 be the ideals in O 2 given by J 1 = x 4 , y 4 and J 2 = x 6 , x 2 y 2 , y 6 .…”

“…. , p. By Proposition 3.5 of [4] we have J ai = J bi i , where J is the ideal of O n generated by the monomials x k such that k ∈ Γ + . In particular, we have (J ai ) I = (J bi i ) I .…”