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<p>In this paper, we study normal forms of analytic saddle-nodes in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb C^{n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">C</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> with any Poincaré rank <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k\in \mathbb N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(k=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order <i>k</i>. One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.</p>

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On k-Summable Normal Forms of Vector Fields with One Zero Eigenvalue

  • 期刊论文

  • 发表日期：

    2025年12月15日

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  • Peter De Maesschalck,
  • Kristian Uldall Kristiansen

摘要

In this paper, we study normal forms of analytic saddle-nodes in \(\mathbb C^{n+1}\) C n + 1 with any Poincaré rank \(k\in \mathbb N\) k ∈ N . The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered \(k=1\) k = 1 . In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order k. One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.