We consider a definable map germ <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(f,g):(\mathbb {R}^n,0) \rightarrow (\mathbb {R}^2,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mrow> <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> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under some conditions on f and g, we establish Lê-Greuel-type formulas; namely, we relate the following quantities: <Equation ID="Equ2"> <EquationSource Format="TEX">$\begin{aligned} & \chi \left( \{f=\alpha \} \cap \{g=\delta \} \cap B_\varepsilon ^n \right) ,\\ & \chi \left( \{f=\alpha \} \cap \{g\ge \delta \} \cap B_\varepsilon ^n \right) -\chi \left( \{f=\alpha \} \cap \{g\le \delta \} \cap B_\varepsilon ^n \right) , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd /> <mtd columnalign="left"> <mrow> <mi>χ</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">{</mo> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <mrow> <mo stretchy="false">{</mo> <mi>g</mi> <mo>=</mo> <mi>δ</mi> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <msubsup> <mi>B</mi> <mi>ε</mi> <mi>n</mi> </msubsup> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow /> </mtd> <mtd columnalign="left"> <mrow> <mi>χ</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">{</mo> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <mrow> <mo stretchy="false">{</mo> <mi>g</mi> <mo>≥</mo> <mi>δ</mi> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <msubsup> <mi>B</mi> <mi>ε</mi> <mi>n</mi> </msubsup> </mfenced> <mo>-</mo> <mi>χ</mi> <mfenced close=")" open="("> <mrow> <mo stretchy="false">{</mo> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <mrow> <mo stretchy="false">{</mo> <mi>g</mi> <mo>≤</mo> <mi>δ</mi> <mo stretchy="false">}</mo> </mrow> <mo>∩</mo> <msubsup> <mi>B</mi> <mi>ε</mi> <mi>n</mi> </msubsup> </mfenced> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\alpha ,\delta )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a regular value of (f, g) and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0< \vert (\alpha ,\delta ) \vert \ll \varepsilon \ll 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mo stretchy="false">|</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">|</mo> <mo>≪</mo> <mi>ε</mi> <mo>≪</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, to Poincaré–Hopf indices of appropriate vector fields. Our method is based on Lagrange multipliers and some polar techniques. We provide a large class of weighted homogeneous mappings for which our conditions are satisfied.

Lê-Greuel-type formulas for some mappings from $(\mathbb {R}^n,0)$ to $(\mathbb {R}^2,0)$

期刊论文
发表日期：

2025年11月17日

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Nicolas Dutertre,
Juan Antonio Moya Pérez

摘要

We consider a definable map germ $(f,g):(\mathbb {R}^n,0) \rightarrow (\mathbb {R}^2,0)$ $(f, g) : (R^{n}, 0) \to (R^{2}, 0)$ . Under some conditions on f and g, we establish Lê-Greuel-type formulas; namely, we relate the following quantities: $\begin{aligned} & \chi \left( \{f=\alpha \} \cap \{g=\delta \} \cap B_\varepsilon ^n \right) ,\\ & \chi \left( \{f=\alpha \} \cap \{g\ge \delta \} \cap B_\varepsilon ^n \right) -\chi \left( \{f=\alpha \} \cap \{g\le \delta \} \cap B_\varepsilon ^n \right) , \end{aligned}$ $\begin{matrix} χ ({f = α} \cap {g = δ} \cap B_{ε}^{n}), \\ χ ({f = α} \cap {g \geq δ} \cap B_{ε}^{n}) - χ ({f = α} \cap {g \leq δ} \cap B_{ε}^{n}), \end{matrix}$ where $(\alpha ,\delta )$ $(α, δ)$ is a regular value of (f, g) and $0< \vert (\alpha ,\delta ) \vert \ll \varepsilon \ll 1$ $0 < | (α, δ) | ≪ ε ≪ 1$ , to Poincaré–Hopf indices of appropriate vector fields. Our method is based on Lagrange multipliers and some polar techniques. We provide a large class of weighted homogeneous mappings for which our conditions are satisfied.

Lê-Greuel-type formulas for some mappings from \((\mathbb {R}^n,0)\) to \((\mathbb {R}^2,0)\)

摘要