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<p>We study corank one <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathscr {A}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation>-finite germs <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(f:(\mathbb {R}^n,0)\rightarrow (\mathbb {R}^{n+1},0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</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> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and their complexifications. More precisely, we study when these germs provide good real pictures of the complex germs, i.e., when there is a real deformation that has the same homology in the image (hence, homotopy) as the generic complex deformation. We give a new sufficient condition that can be computed in practice, as well as examples.</p>

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Good real pictures of corank one map germs from the n-space to the \((n+1)\)-space

  • 期刊论文

  • 发表日期：

    2025年12月3日

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  • I. Breva Ribes,
  • R. Giménez Conejero

摘要

We study corank one \(\mathscr {A}\) A -finite germs \(f:(\mathbb {R}^n,0)\rightarrow (\mathbb {R}^{n+1},0)\) f : ( R n , 0 ) → ( R n + 1 , 0 ) and their complexifications. More precisely, we study when these germs provide good real pictures of the complex germs, i.e., when there is a real deformation that has the same homology in the image (hence, homotopy) as the generic complex deformation. We give a new sufficient condition that can be computed in practice, as well as examples.