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<p>A <i>cubic space</i> is a vector space equipped with a symmetric trilinear form. Two cubic spaces are <i>isogeneous</i> if each embeds into the other. A cubic space is <i>non-degenerate</i> if its form cannot be expressed as a finite sum of products of linear and quadratic forms. We classify non-degenerate cubic spaces of countable dimension up to isogeny: the isogeny classes are completely determined by an invariant we call the <i>residual rank</i>, which takes values in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1056_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textbf{N}\cup \{\infty \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">N</mi> <mo>∪</mo> <mo stretchy="false">{</mo> <mi>∞</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, the set of classes is discrete and (under the partial order of embedability) satisfies the descending chain condition.</p>

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Isogeny classes of cubic spaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月31日

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  • Arthur Bik,
  • Alessandro Danelon,
  • Andrew Snowden

摘要

A cubic space is a vector space equipped with a symmetric trilinear form. Two cubic spaces are isogeneous if each embeds into the other. A cubic space is non-degenerate if its form cannot be expressed as a finite sum of products of linear and quadratic forms. We classify non-degenerate cubic spaces of countable dimension up to isogeny: the isogeny classes are completely determined by an invariant we call the residual rank, which takes values in \(\textbf{N}\cup \{\infty \}\) N ∪ { ∞ } . In particular, the set of classes is discrete and (under the partial order of embedability) satisfies the descending chain condition.