Joint differential resolvents for pseudopolynomials

TL;DR

This paper extends the concept of differential resolvents from single monomials to pseudopolynomials involving multiple roots and generalizes the resolvent to include non-consecutive derivatives, providing a computational method.

Contribution

It introduces a generalized framework for differential resolvents applicable to pseudopolynomials and multiple roots, expanding previous single-polynomial results.

Findings

Generalized differential resolvent for pseudopolynomials

Use of powersum formula for computation

Extension to non-consecutive derivatives

Abstract

The existence of linear differential resolvents for z^alpha for any root z of an ordinary polynomial with coefficients in a given ordinary differential field has been established, where alpha is an indeterminate constant with respect to the derivation of the given field. In this paper we consider several alphas. We will call a finite sum of indeterminate powers of a variable v a pseudopolynomial in v. We will generalize the definition of a differential resolvent of a single polynomial for a single monomial z^alpha to the definition of a differential resolvent of several polynomials for a pseudopolynomial in the roots. We will also generalize the definition of a resolvent to have non-consecutive derivatives. We will show that the authors powersum formula may be used to compute this more general differential resolvent.