Invariants for bi-Lipschitz equivalence of ideals

Abstract

We introduce the notion of bi-Lipschitz equivalence of ideals and derive numerical invariants for such equivalence. In particular, we show that the log canonical threshold of ideals is a bi-Lipschitz invariant. We apply our method to several deformations f t : (C n , 0) → (C, 0) and show that they are not bi-Lipschitz trivial, specially focusing on several known examples of non µ *-constant deformations.

The sequence of mixed Łojasiewicz exponents associated to pairs of monomial ideals

“…, e n (I, J) when I is Hickel with respect to J. The following result is analogous to [10,Lemma 5.5]. Thus (24) follows.…”

“…Due to the fundamental work of Lejeune and Teissier [17], Lojasiewicz exponents admit an equivalent formulation in terms of the notion of integral closure of ideals. Consequently, these numbers have a translation in terms of multiplicities of ideals, by virtue of the Rees' multiplicity theorem (see relation 10).…”

Bi-Lipschitz contact invariance of rank

“…Results in these lemmas are well known and easy to prove. One might find an analogous statement of Lemma 2.1 for function germs in [1] and a proof of Lemma 2.2 in [3], Theorem 3.2. For convenience, we provide here with proofs.…”

Classification of Lipschitz simple function germs

“…Let g be a corank 4 germ. By the Splitting lemma, we may assume (up to addition of a non-degenerate quadratic form of further variables) that g ∈ m 3 4 . Consider a small deformation of g of the form h λ = g + λ(x 3 + y 3 + z 3 + w 3 ).…”

“…The only pair left in our analysis is (Q 11 , S 11 ). This pair is shown to be of different bi-Lipschitz type by using a result of Bivià-Ausina and Fukui [3] on the bi-Lipschitz invariance of the log canonical threshold of ideals.…”