Discussion on polynomials having polynomial iterative roots

  • Authors:

  • Zhiheng Yu;Lu Yang;Weinian Zhang

  • Affiliations:

  • Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China;Shanghai Key Laboratory for Trustworthy Computing, East China Normal University, Shanghai 200062, PR China;Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2012

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Abstract

In this paper, we discuss polynomial mappings which have iterative roots of the polynomial form. We apply the computer algebra system Singular to decompose algebraic varieties and finally find a condition under which polynomial functions have quadratic iterative roots of quadratic polynomial form. This condition is equivalent to, but simpler than Schweizer and Sklar's and more convenient than Strycharz-Szemberg and Szemberg's. We further find all polynomial functions which have cubic iterative roots of the quadratic polynomial form and compute all those iterative roots. Moreover, we find all 2-dimensional homogeneous polynomial mappings of degree 2 which have iterative roots of the polynomial form and obtain expressions of some iterative roots.