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A partial factorization of the powersum formula

International Journal of Mathematics and Mathematical Sciences
Publication Date
  • Mathematics


For any univariate polynomial P whose coefficients lie in an ordinary differential field 𝔽 of characteristic zero, and for any constant indeterminate α, there exists a nonunique nonzero linear ordinary differential operator ℜ of finite order such that the αth power of each root of P is a solution of ℜzα=0, and the coefficient functions of ℜ all lie in the differential ring generated by the coefficients of P and the integers ℤ. We call ℜ an α-resolvent of P. The author's powersum formula yields one particular α-resolvent. However, this formula yields extremely large polynomials in the coefficients of P and their derivatives. We will use the A-hypergeometric linear partial differential equations of Mayr and Gelfand to find a particular factor of some terms of this α-resolvent. We will then demonstrate this factorization on an α-resolvent for quadratic and cubic polynomials.

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