<p>We investigate sufficient conditions for the invariance of the real Milnor number under <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ \mathcal {R} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>-bi-Lipschitz equivalence for function germs <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ f, g :(\mathbb {R}^n, 0) \rightarrow (\mathbb {R}, 0) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. More generally, we explore its invariance within the extended framework of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$ \mathcal {R} $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>-asymptotically Lipschitz equivalence. To this end, we introduce the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-derivative of maps, which provides a natural setting for studying asymptotic growth. Additionally, we discuss the implications of our results in the context of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ C^k $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ C^{\infty } $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-equivalences, establishing sufficient conditions for the real Milnor number to remain invariant.</p>

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

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2026年2月28日

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Raphael de Omena,
José Edson Sampaio,
Emanoel Souza

摘要

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