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<p>We establish effective bounds on the number of periodic points of polynomials <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> defined over <i>p</i>-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(X^d + c\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>X</mi> <mi>d</mi> </msup> <mo>+</mo> <mi>c</mi> </mrow> </math></EquationSource> </InlineEquation> with <i>c</i> integral at some prime dividing <i>d</i>. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields <i>K</i>, giving the explicit uniform bound <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\#\textrm{Per}_K(\phi ) \le d^{[K:\mathbb {Q}]}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>#</mo> <msub> <mtext>Per</mtext> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msup> <mi>d</mi> <mrow> <mo stretchy="false">[</mo> <mi>K</mi> <mo>:</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Uniform bounds on periodic points of polynomials with good reduction

  • 期刊论文

  • 发表日期：

    2026年7月22日

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  • Isaac Rajagopal,
  • Robin Zhang

摘要

We establish effective bounds on the number of periodic points of polynomials \(\phi \) ϕ defined over p-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials \(X^d + c\) X d + c with c integral at some prime dividing d. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields K, giving the explicit uniform bound \(\#\textrm{Per}_K(\phi ) \le d^{[K:\mathbb {Q}]}\) # Per K ( ϕ ) ≤ d [ K : Q ] .