This article gives a recursion for the minimal generators <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">$G = \{g_i\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>=</mo> <mo stretchy="false">{</mo> <msub> <mi>g</mi> <mi>i</mi> </msub> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> of the generic value set <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\Lambda _{\text {gen}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mtext>gen</mtext> </msub> </math></EquationSource> </InlineEquation> of a plane curve germ C with a two-generator semigroup <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="81" /> </InlineMediaObject> <EquationSource Format="TEX">$\Gamma = \langle p, m \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Γ</mi> <mo>=</mo> <mo stretchy="false">⟨</mo> <mi>p</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation>. The main result provides a recursive formula producing all <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$g_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>g</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>, and shows that they are in fact minimal generators. From this recursion we deduce a formula for the cardinality of the minimal generators |G| and for the conductor <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$c(\Lambda _{\text {gen}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">Λ</mi> <mtext>gen</mtext> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">$\Lambda _{\text {gen}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Λ</mi> <mtext>gen</mtext> </msub> </math></EquationSource> </InlineEquation>. The recursion can be used to compute the generic Tjurina number <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10569_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau _{\text {gen}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>τ</mi> <mtext>gen</mtext> </msub> </math></EquationSource> </InlineEquation>, a method compared to that given by M. Alberich-Carramiñana et al. (Indiana Univ. Math. J. 70(4), 1211–1220 (2021)). The main result is based on the algorithm and ideas of C. Delorme (C. R. Acad. Sci. Paris Sér. A 279, 367–369 (1974); Bull. Soc. Math. France 106, 417–446 (1978)).

The generic value set of a two-generator semigroup

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2025年8月18日

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Justin Lake,
Lee McEwan

摘要

This article gives a recursion for the minimal generators $G = \{g_i\}$ $G = {g_{i}}$ of the generic value set $\Lambda _{\text {gen}}$ $Λ_{gen}$ of a plane curve germ C with a two-generator semigroup $\Gamma = \langle p, m \rangle $ $Γ = ⟨ p, m ⟩$ . The main result provides a recursive formula producing all $g_i$ $g_{i}$ , and shows that they are in fact minimal generators. From this recursion we deduce a formula for the cardinality of the minimal generators |G| and for the conductor $c(\Lambda _{\text {gen}})$ $c (Λ_{gen})$ of $\Lambda _{\text {gen}}$ $Λ_{gen}$ . The recursion can be used to compute the generic Tjurina number $\tau _{\text {gen}}$ $τ_{gen}$ , a method compared to that given by M. Alberich-Carramiñana et al. (Indiana Univ. Math. J. 70(4), 1211–1220 (2021)). The main result is based on the algorithm and ideas of C. Delorme (C. R. Acad. Sci. Paris Sér. A 279, 367–369 (1974); Bull. Soc. Math. France 106, 417–446 (1978)).