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<p>In this work, we provide a concise proof of the logarithmic derivative lemma in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1701_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\({\textbf{C}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold">C</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. We start by giving a complete proof of a critical result by Biancofiore and Stoll (Ann. of Math. Stud., No. 100, 29–45, 1981) via matrix theory, which is central to this area of study. Then, we follow Li (Trans. Amer. Math. Soc. 363, 6257–6267, 2011) and Ye (Math. Z. 222, 81–95, 1996) to present the shortest proof to date.</p>

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A Short Proof of the Lemma of the Logarithmic Derivative in Several Complex Variables

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  • 发表日期：

    2025年5月28日

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  • Qi Han,
  • Jingbo Liu

摘要

In this work, we provide a concise proof of the logarithmic derivative lemma in \({\textbf{C}}^n\) C n . We start by giving a complete proof of a critical result by Biancofiore and Stoll (Ann. of Math. Stud., No. 100, 29–45, 1981) via matrix theory, which is central to this area of study. Then, we follow Li (Trans. Amer. Math. Soc. 363, 6257–6267, 2011) and Ye (Math. Z. 222, 81–95, 1996) to present the shortest proof to date.