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<p>We explore distribution questions for rational maps on the projective line <InlineEquation ID="IEq3"> <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> over <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {Q}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation> within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\phi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϕ</mi> </math></EquationSource> </InlineEquation> of fixed degree <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(d \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> with prescribed reduction properties. One of our main results establishes that with respect to the weak box densities in an earlier work of Poonen, a positive proportion of rational maps consist of those having globally minimal resultant. Additionally, for degree 2 rational maps, we perform explicit computations demonstrating that over <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(32.7\%\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>32.7</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> possess a squarefree, and hence minimal, resultant.</p>

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Counting rational maps on \(\mathbb {P}^1\) with prescribed local conditions

  • 期刊论文

  • 发表日期：

    2026年1月28日

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  • Khoa D. Nguyen,
  • Anwesh Ray

摘要

We explore distribution questions for rational maps on the projective line \(\mathbb {P}^1\) P 1 over \(\mathbb {Q}\) Q within the framework of arithmetic dynamics, drawing analogies to elliptic curves. Specifically, we investigate counting problems for rational maps \(\phi \) ϕ of fixed degree \(d \ge 2\) d ≥ 2 with prescribed reduction properties. One of our main results establishes that with respect to the weak box densities in an earlier work of Poonen, a positive proportion of rational maps consist of those having globally minimal resultant. Additionally, for degree 2 rational maps, we perform explicit computations demonstrating that over \(32.7\%\) 32.7 % possess a squarefree, and hence minimal, resultant.