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Computer Mathematics pp 125–140Cite as

Series-Expansion of Multivariate Algebraic Functions at Singular Points: Nonmonic Case

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Abstract

In a series of papers, we have developed a method of expanding multivariate algebraic functions at their singular points. The method applies the Hensel construction to the defining polynomial of the algebraic function, so we named the resulting series “Hensel series”. In [1], we derived a concise representation of Hensel series for the monic defining polynomial, and clarified several characteristic properties of Hensel series theoretically. In this paper, we study the case of nonmonic defining polynomial. We show that, by determining the so-called Newton polynomial suitably, we can construct Hensel series which show reasonable behaviors at zero-points of the leading coefficients and we can derive a representation of Hensel series in the nonmonic case just similarly as in the monic case. Furthermore, we investigate the convergence/divergence behavior and many-valuedness of Hensel series in the nonmonic case.

Work supported in part by Japan Society for the Promotion of Science under Grants 19300001.

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Author information

Authors and Affiliations

  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Japan

    Tateaki Sasaki

  2. Mathematics Certification Institute of Japan, Katsushika-ku, Japan

    Daiju Inaba

Authors
  1. Tateaki Sasaki
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  2. Daiju Inaba
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Corresponding authors

Correspondence to Tateaki Sasaki or Daiju Inaba .

Editor information

Editors and Affiliations

  1. Academy of Mathematics & System Science, Beijing, China

    Ruyong Feng

  2. Department of Mathematics and Computer S, University of Antwerp, Antwerp, Antwerpen, Belgium

    Wen-shin Lee

  3. Department of Mathematical Information S, Tokyo University of Science, Tokyo, Tokyo, Japan

    Yosuke Sato

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Sasaki, T., Inaba, D. (2014). Series-Expansion of Multivariate Algebraic Functions at Singular Points: Nonmonic Case. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_11

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