In this study, we examine the Saito number of a plane curve and propose a method for determining the minimum Saito number within a given equisingularity class, resulting in an actual algorithm. In certain cases, we also offer explicit formulas for this number. In addition, if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_465_Article_IEq1.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu _0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ν</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_465_Article_IEq2.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ν</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> are coprime integers with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_465_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="84" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le \nu _0<\nu _1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <msub> <mi>ν</mi> <mn>0</mn> </msub> <mo><</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_465_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$N>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we show that there exists a plane curve equisingular to the curve <Equation ID="Equ11"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_465_Article_Equ11.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} y^{N\nu _0}-x^{N\nu _1}=0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>y</mi> <mrow> <mi>N</mi> <msub> <mi>ν</mi> <mn>0</mn> </msub> </mrow> </msup> <mo>-</mo> <msup> <mi>x</mi> <mrow> <mi>N</mi> <msub> <mi>ν</mi> <mn>1</mn> </msub> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>such that its Saito number is equal to k, for any <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="574_2025_465_Article_IEq5.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="113" /> </InlineMediaObject> <EquationSource Format="TEX">$1\le k\le \left[ \frac{N\nu _0}{2}\right] .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mfenced close="]" open="["> <mfrac> <mrow> <mi>N</mi> <msub> <mi>ν</mi> <mn>0</mn> </msub> </mrow> <mn>2</mn> </mfrac> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

On the Saito Number of Plane Curves

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2025年7月4日

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Yohann Genzmer,
Marcelo Escudeiro Hernandes

摘要

In this study, we examine the Saito number of a plane curve and propose a method for determining the minimum Saito number within a given equisingularity class, resulting in an actual algorithm. In certain cases, we also offer explicit formulas for this number. In addition, if $\nu _0$ $ν_{0}$ and $\nu _1$ $ν_{1}$ are coprime integers with $1\le \nu _0<\nu _1$ $1 \leq ν_{0} < ν_{1}$ and $N>0$ $N > 0$ , we show that there exists a plane curve equisingular to the curve $\begin{aligned} y^{N\nu _0}-x^{N\nu _1}=0 \end{aligned}$ $\begin{matrix} y^{N ν_{0}} - x^{N ν_{1}} = 0 \end{matrix}$ such that its Saito number is equal to k, for any $1\le k\le \left[ \frac{N\nu _0}{2}\right] .$ $1 \leq k \leq [\frac{N ν_{0}}{2}] .$