A numerical study of extended Hensel series

ABSTRACT

The extended Hensel construction is a Hensel construction at a singular point of the multivariate polynomial, and it allows us to expand the roots of a given multivariate poly-nomial into a kind of series which we call an extended Hensel series. This paper investigates the behavior of the extended Hensel series numerically, and clarifies the following four points. 1) The convergence domain of the extended Hensel series is very different from those of the Taylor series; both convergence and divergence domains coexist in the neighborhood of the expansion point. 2) The extended Hensel series truncated at 7 ~ 8 order coincides very well with the corresponding algebraic function in the convergence domain, while it behaves very wildly in the divergence domain. 3) In the case of non-monic polynomial, the factors of leading co-efficient are distributed among the extended Hensel series, and the singular behaviors of the roots at the zero-points of the leading coefficient are expressed nicely by the Hensel series. 4) Although many-valuedness of extended Hensel series is usually different from that of the corresponding exact roots, the Hensel series reproduce the behaviors of the exact roots by jumping from one branch to another occasionally.

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Index Terms

• Computing methodologies

  • Symbolic and algebraic manipulation

    • Symbolic and algebraic algorithms

• Mathematics of computing

  • Discrete mathematics

    • Graph theory

      • Approximation algorithms

  • Mathematical analysis

    • Functional analysis

      • Approximation

    • Nonlinear equations

• Theory of computation

  • Design and analysis of algorithms

    • Approximation algorithms analysis