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<p>This paper is devoted to a class of complete hyperelliptic Abelian integrals of the first kind, and aims to investigate a related conjecture, proposed by Gavrilov and Iliev (2003, Trans. Amer. Math. Soc., 1185-1207), that for some domain of the parameters, the corresponding hyperelliptic Abelian integrals satisfy the Chebyshev property with accuracy two (i.e., they have at most <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{3}\)</EquationSource> </InlineEquation> zeros counted with multiplicity). We show that, however, for a subset of the above domain, the corresponding integrals satisfy the Chebyshev property, which gives an answer to the above conjecture.</p>

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On the Number of Zeros of Some Complete Hyperelliptic Abelian Integrals of the First Kind

  • 期刊论文

  • 发表日期：

    2025年10月27日

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  • Yangjian Sun,
  • Shaoqing Wang,
  • Jiazhong Yang

摘要

This paper is devoted to a class of complete hyperelliptic Abelian integrals of the first kind, and aims to investigate a related conjecture, proposed by Gavrilov and Iliev (2003, Trans. Amer. Math. Soc., 1185-1207), that for some domain of the parameters, the corresponding hyperelliptic Abelian integrals satisfy the Chebyshev property with accuracy two (i.e., they have at most \(\varvec{3}\) zeros counted with multiplicity). We show that, however, for a subset of the above domain, the corresponding integrals satisfy the Chebyshev property, which gives an answer to the above conjecture.