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The real Milnor number remains invariant under asymptotically Lipschitz equivalence for function germs. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

March 2, 2026Research in the Mathematical Sciences0 citationsOpen Access

On the invariance of the real milnor number under asymptotically Lipschitz equivalence

RORaphael de OmenaUniversidade Federal do Ceará JSJosé Edson SampaioUniversidade Federal do Ceará ESEmanoel SouzaUniversidade Estadual do Ceará

Key Points

The aim is to investigate the conditions for the invariance of the real Milnor number under various equivalences.
Analyzed function germs mapping from R^n to R.
Introduced the alpha-derivative for studying asymptotic growth.
Explored extended frameworks for asymptotic Lipschitz equivalence.
Established sufficient conditions for the invariance of the real Milnor number.
Demonstrated implications for C^k and C^{ ext{∞}} equivalences.

Abstract

Abstract We investigate sufficient conditions for the invariance of the real Milnor number under R R -bi-Lipschitz equivalence for function germs f, g: (Rⁿ, 0) (R, 0) f, g: (R n, 0) → (R, 0). More generally, we explore its invariance within the extended framework of R R -asymptotically Lipschitz equivalence. To this end, we introduce the α -derivative of maps, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of Cᵏ C k and C^ C ∞ -equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.

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