[Paper Review] Semicontinuity of the {\L}ojasiewicz exponent

Arkadiusz Płoski|arXiv (Cornell University)|Feb 23, 2011

Holomorphic and Operator Theory4 references5 citations

TL;DR

This paper establishes the lower semicontinuity of the Łojasiewicz exponent $ l_0(f) $ for finite holomorphic germs $ f: (ℂ^n,0) \to (ℂ^n,0) $ under multiplicity-constant deformations. Using the characteristic polynomial of a function relative to the germ and properties of Newton polygons, it proves that $ l_0(F_0) \leq l_0(F_t) $ for $ t $ near 0, with equality in the one-parameter case. The result extends Teissier's semicontinuity for $ \mu $-constant deformations to the broader class of multiplicity-constant deformations.

ABSTRACT

We prove that the {\\L}ojasiewicz exponent $l_0(f)$ of a finite holomorphic germ $f:(C^n,0)\ o(C^n,0)$ is lower semicontinuous in any multiplicity-constant deformation of $f$.

Motivation & Objective

• To establish the semicontinuity of the Łojasiewicz exponent $ l_0(f) $ for finite holomorphic germs under multiplicity-constant deformations.
• To generalize Teissier's earlier result on $ \mu $-constant deformations to the broader class of multiplicity-constant deformations.
• To analyze the behavior of the Łojasiewicz exponent via the characteristic polynomial and Newton polygon of a function relative to the germ.
• To show that the Newton polygon of the function relative to the deformed germ is contained in that of the original germ, reflecting semicontinuity.

Proposed method

• Uses the definition of the Łojasiewicz exponent as $ l_0(f) = \sup_{\phi} \left\{ \frac{\mathrm{ord}(f \circ \phi)}{\mathrm{ord}(\phi)} \right\} $ over analytic arcs $ \phi $.
• Applies the characteristic polynomial $ P_h(t,w,s) \in \mathbb{C}\{t,w\}[s] $ of a function $ h $ relative to the deformation $ F(t,z) $, which encodes the algebraic structure of the germ.
• Defines the order $ o_{F_t}(h) = \inf_i \left\{ \frac{\mathrm{ord}(a_i(t,w))}{i} \right\} $, showing that this order is semicontinuous in $ t $.
• Uses the Newton polygon $ \mathcal{N}(F_t, h) $ of the characteristic polynomial to analyze the Łojasiewicz exponent, with the inclination of the first edge giving $ 1/o_{F_t}(h) $.
• Proves that $ \mathcal{N}(F_t, h) \subset \mathcal{N}(F_0, h) $ for $ t $ near 0, implying semicontinuity of the order and thus of the Łojasiewicz exponent.
• Establishes that in the one-parameter case, $ o_{F_t}(h) $ is constant for $ t \neq 0 $, hence $ l_0(F_t) $ is constant.

Experimental results

Research questions

• RQ1 Is the Łojasiewicz exponent lower semicontinuous under multiplicity-constant deformations of finite holomorphic germs?
• RQ2 How does the Newton polygon of a function relative to a germ behave under such deformations?
• RQ3 Can the semicontinuity of the Łojasiewicz exponent be established via the characteristic polynomial of the function relative to the deformation?
• RQ4 Does the one-parameter case yield constancy of the Łojasiewicz exponent away from the central fiber?
• RQ5 What is the relationship between the Newton polygon of the deformed germ and that of the original germ?

Key findings

• The Łojasiewicz exponent $ l_0(f) $ is lower semicontinuous under multiplicity-constant deformations: $ l_0(F_0) \leq l_0(F_t) $ for $ t $ near 0.
• In the one-parameter case, $ l_0(F_t) $ is constant for $ t \neq 0 $, implying no jump occurs away from the central fiber.
• The Newton polygon $ \mathcal{N}(F_t, h) $ of the function $ h $ relative to $ F_t $ satisfies $ \mathcal{N}(F_t, h) \subset \mathcal{N}(F_0, h) $ for $ t $ near 0.
• The order $ o_{F_t}(h) $, which determines $ l_0(F_t) $, is semicontinuous in $ t $, with $ o_{F_t}(h) \leq o_{F_0}(h) $.
• The Łojasiewicz exponent $ l_0(f) $ satisfies $ l_0(f) \leq \mathrm{m}_0(f) $, with equality iff the Jacobian has rank at least $ n-1 $ at 0.
• The Newton polygon $ \mathcal{N}(f,h) $ coincides with Teissier's $ \mathcal{N}_{\mathrm{I}(f)}(h) $, linking the result to existing theory on mixed multiplicities.