Let <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_624_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$F, G \in \mathbb {Z}[X, Y]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>,</mo> <mi>G</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> be binary forms of degree <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_624_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, non-zero discriminant and with automorphism group isomorphic to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_624_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$D_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_624_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="109" /> </InlineMediaObject> <EquationSource Format="TEX">$F(\mathbb {Z}^2) = G(\mathbb {Z}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we show that F and G are <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_624_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">${\textrm{GL}}(2, \mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>GL</mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>–equivalent.

Binary forms with the same value set II: The case of ${\textbf {D}}_4$

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2025年3月15日

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Étienne Fouvry,
Peter Koymans

摘要

Let $F, G \in \mathbb {Z}[X, Y]$ $F, G \in Z [X, Y]$ be binary forms of degree $\ge 3$ $\geq 3$ , non-zero discriminant and with automorphism group isomorphic to $D_4$ $D_{4}$ . If $F(\mathbb {Z}^2) = G(\mathbb {Z}^2)$ $F (Z^{2}) = G (Z^{2})$ , we show that F and G are ${\textrm{GL}}(2, \mathbb {Z})$ $GL (2, Z)$ –equivalent.

Binary forms with the same value set II: The case of \({\textbf {D}}_4\)

摘要