This paper is devoted to the number of isolated zeros of the hyperelliptic integrals<Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2023_2334_Article_Equ1.gif" Format="GIF" Height="44" Rendition="HTML" Resolution="72" Type="Linedraw" Width="258" /> </MediaObject> <EquationSource Format="TEX">$I(h) = \oint_{{\Gamma _h}} {({\alpha + \beta x})ydx,\;\;\; \alpha ,\beta \in \mathbb{R},}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∮</mo> <mrow> <mrow> <msub> <mi mathvariant="normal">Γ</mi> <mi>h</mi> </msub> </mrow> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mi>x</mi> </mrow> <mo stretchy="false">)</mo> <mi>y</mi> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where Γh is a compact component of the hyperelliptic cures {(x,y) ∣ y2 + P5(x) = h, h ∈ Σ}; here, Σ is a maximal open interval on which a continuous family of ovals Γh exists, and P5(x) is a polynomial of x with degree five. As is shown in Liu and Xiao (2013), P5(x) can be assumed to have the form<Equation ID="Equ2"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2023_2334_Article_Equ2.gif" Format="GIF" Height="36" Rendition="HTML" Resolution="72" Type="Linedraw" Width="385" /> </MediaObject> <EquationSource Format="TEX">${P_5}(x) = - {{uv} \over 2}{x^2} + {{u + v + uv} \over 3}{x^3} - {{1 + u + v} \over 4}{x^4} + {1 \over 5}{x^5},$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>P</mi> <mn>5</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow> <mfrac> <mrow> <mi>u</mi> <mi>v</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <mfrac> <mrow> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>+</mo> <mi>u</mi> <mi>v</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mrow> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mrow> <msup> <mi>x</mi> <mn>5</mn> </msup> </mrow> <mo>,</mo> </math></EquationSource> </Equation>and there exist some real numbers α and β such that I(h) has at least two isolated zeros if (v, u) ∈ Θ, where<Equation ID="Equ3"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2023_2334_Article_Equ3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="467" /> </MediaObject> <EquationSource Format="TEX">$\Theta=\{(v,u):v=\bar{u},\;\text{Im}(v)\ne0\;\text{and}\;(\text{Re}(u)-2)^{2}+(\text{Im}(v))^{2}<1\}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi mathvariant="normal">Θ</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>v</mi> <mo>,</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>v</mi> <mo>=</mo> <mrow> <mover> <mi>u</mi> <mo stretchy="false">¯</mo> </mover> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mtext>Im</mtext> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mtext>and</mtext> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mtext>Re</mtext> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mtext>Im</mtext> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow> <mn>2</mn> </mrow> </msup> <mo><</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </math></EquationSource> </Equation>However, the problem whether two is also the upper bound of the number of isolated zeros of I(h) remains open for (v,u) ∈ Θ. In this paper, we propose a new simplification technique, which can reduce the degree of a polynomial by at least half. This combined with the new criterion and some available methods and techniques shows that two is indeed the lowest upper bound on the number of isolated zeros of I(h) for some subset of Θ, which partially gives a positive answer to the open problem. The methods and techniques developed in this paper may be used to study other similar problems.

The lowest upper bound on the number of zeros of a class of hyperelliptic Abelian integrals

期刊论文
发表日期：

2025年1月22日

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Yangjian Sun,
Yanfei Dai

摘要

This paper is devoted to the number of isolated zeros of the hyperelliptic integrals

$I(h) = \oint_{{\Gamma _h}} {({\alpha + \beta x})ydx,\;\;\; \alpha ,\beta \in \mathbb{R},}$ $I (h) = \oint_{Γ_{h}} (α + β x) y d x, α, β \in R,$

where Γ_h is a compact component of the hyperelliptic cures {(x,y) ∣ y² + P₅(x) = h, h ∈ Σ}; here, Σ is a maximal open interval on which a continuous family of ovals Γ_h exists, and P₅(x) is a polynomial of x with degree five. As is shown in Liu and Xiao (2013), P₅(x) can be assumed to have the form

${P_5}(x) = - {{uv} \over 2}{x^2} + {{u + v + uv} \over 3}{x^3} - {{1 + u + v} \over 4}{x^4} + {1 \over 5}{x^5},$ $P_{5} (x) = - \frac{u v}{2} x^{2} + \frac{u + v + u v}{3} x^{3} - \frac{1 + u + v}{4} x^{4} + \frac{1}{5} x^{5},$

and there exist some real numbers α and β such that I(h) has at least two isolated zeros if (v, u) ∈ Θ, where

$\Theta=\{(v,u):v=\bar{u},\;\text{Im}(v)\ne0\;\text{and}\;(\text{Re}(u)-2)^{2}+(\text{Im}(v))^{2}<1\}.$ $Θ = {(v, u) : v = \bar{u}, Im (v) \neq 0 and (Re (u) - 2)^{2} + (Im (v))^{2} < 1} .$

However, the problem whether two is also the upper bound of the number of isolated zeros of I(h) remains open for (v,u) ∈ Θ. In this paper, we propose a new simplification technique, which can reduce the degree of a polynomial by at least half. This combined with the new criterion and some available methods and techniques shows that two is indeed the lowest upper bound on the number of isolated zeros of I(h) for some subset of Θ, which partially gives a positive answer to the open problem. The methods and techniques developed in this paper may be used to study other similar problems.