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Abstract <p> We describe properties of 2-adic valuation trees associated to depressed cubic polynomials. By doing so, we provide a criterion for the solvability of a depressed cubic equation and the number of solutions over some domains of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1114_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{Q}_2\)</EquationSource> </InlineEquation>. This leads to the locality of the solutions of the depressed cubic equation over <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1114_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{Q}_2\)</EquationSource> </InlineEquation>. We also present an example illustrating how the dynamics of depressed cubic polynomials with infinite tree branches may fail to be topologically transitive (i.e., dense) over the set <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12607_2025_1114_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{Z}_2\)</EquationSource> </InlineEquation>. </p>

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Trees and Solvability of Depressed Cubics over \(\mathbb{Q}_{2}\)

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • M. A. K. Ahmad,
  • M. Alp,
  • O. Kozhushkina,
  • J. Long,
  • M. Saburov,
  • J. Trulen

摘要

Abstract

We describe properties of 2-adic valuation trees associated to depressed cubic polynomials. By doing so, we provide a criterion for the solvability of a depressed cubic equation and the number of solutions over some domains of \(\mathbb{Q}_2\) . This leads to the locality of the solutions of the depressed cubic equation over \(\mathbb{Q}_2\) . We also present an example illustrating how the dynamics of depressed cubic polynomials with infinite tree branches may fail to be topologically transitive (i.e., dense) over the set \(\mathbb{Z}_2\) .