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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> be the unit circle in the complex plane and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> be the open unit disc. The Möbius transformations that preserve <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {D}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> are well known and find numerous applications in complex analysis and geometry. We are interested in symmetric multi-affine polynomials <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(P(z_1, \ldots , z_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with the property that for any <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(u_3,\ldots , u_n\in \mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo>∈</mo> <mi mathvariant="script">C</mi> </mrow> </math></EquationSource> </InlineEquation>, there exist <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(u_1, u_2\in \mathcal {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>u</mi> <mn>2</mn> </msub> <mo>∈</mo> <mi mathvariant="script">C</mi> </mrow> </math></EquationSource> </InlineEquation>, such that <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(P(u_1,\ldots ,u_n)=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>u</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We thoroughly investigate and characterize the symmetric multi-affine polynomials with that property, in the cases <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(n=2,3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. When <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(n=2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> this amounts to a characterization of the Möbius transformations <i>T</i> that are idempotents and have the property <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\mathcal {C} \cap T(\mathcal {C}) \not = \emptyset .\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo>∩</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi mathvariant="normal">∅</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

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On the geometry of polynomials acting on the unit circle

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Hristo Sendov

摘要

Let \(\mathcal {C}\) C be the unit circle in the complex plane and let \(\mathbb {D}\) D be the open unit disc. The Möbius transformations that preserve \(\mathbb {D}\) D and \(\mathcal {C}\) C are well known and find numerous applications in complex analysis and geometry. We are interested in symmetric multi-affine polynomials \(P(z_1, \ldots , z_n)\) P ( z 1 , … , z n ) with the property that for any \(u_3,\ldots , u_n\in \mathcal {C}\) u 3 , … , u n ∈ C , there exist \(u_1, u_2\in \mathcal {C}\) u 1 , u 2 ∈ C , such that \(P(u_1,\ldots ,u_n)=0\) P ( u 1 , … , u n ) = 0 . We thoroughly investigate and characterize the symmetric multi-affine polynomials with that property, in the cases \(n=2,3\) n = 2 , 3 . When \(n=2\) n = 2 this amounts to a characterization of the Möbius transformations T that are idempotents and have the property \(\mathcal {C} \cap T(\mathcal {C}) \not = \emptyset .\) C ∩ T ( C ) ≠ ∅ .