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<p>This paper continues discussions in the author’s previous paper about the Misiurewicz polynomials defined for a family of degree <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2069_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(d \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> rational maps with an automorphism group containing the cyclic group of order <i>d</i>. In particular, we extend the sufficient conditions that the Misiurewicz polynomials are irreducible over <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2069_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>. We also prove that the Misiurewicz polynomials always have an irreducible factor of large degree over <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2069_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {Q}_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Q</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="605_2025_2069_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(d=p\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mi>p</mi> </mrow> </math></EquationSource> </InlineEquation> is prime.</p>

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Misiurewicz polynomials for rational maps with nontrivial automorphisms II

  • 期刊论文

  • 发表日期：

    2025年3月21日

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  • Minsik Han

摘要

This paper continues discussions in the author’s previous paper about the Misiurewicz polynomials defined for a family of degree \(d \ge 2\) d ≥ 2 rational maps with an automorphism group containing the cyclic group of order d. In particular, we extend the sufficient conditions that the Misiurewicz polynomials are irreducible over \(\mathbb {Q}\) Q . We also prove that the Misiurewicz polynomials always have an irreducible factor of large degree over \(\mathbb {Q}_p\) Q p when \(d=p\) d = p is prime.