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<p>We establish precise relationships among four distinct positivity conditions for bihomogeneous polynomials. We consider various one-parameter families <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2228_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(r_t\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> of metrics on the line bundle <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2228_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(U^{m}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>U</mi> <mi>m</mi> </msup> </math></EquationSource> </InlineEquation> over projective space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2228_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb P}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and give precise results about how these positivity conditions depend on <i>t</i> and <i>m</i>.</p>

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Positivity Conditions for Hermitian Polynomials

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  • 发表日期：

    2025年10月13日

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  • John P. D’Angelo

摘要

We establish precise relationships among four distinct positivity conditions for bihomogeneous polynomials. We consider various one-parameter families \(r_t\) r t of metrics on the line bundle \(U^{m}\) U m over projective space \({\mathbb P}^1\) P 1 and give precise results about how these positivity conditions depend on t and m.