Powersum formula for polynomials whose distinct roots are differentially independent over constants

We prove that the author's powersum formula yields a nonzero expression for a particular linear ordinary differential equation, called a resolvent, associated with a univariate polynomial whose coefficients lie in a differential field of characteristic zero provided the distinct roots of the polynomial are differentially independent over constants. By definition, the terms of a resolvent lie in the differential field generated by the coefficients of the polynomial, and each of the roots of the polynomial are solutions of the resolvent. One example shows how the powersum formula works. Another example shows how the proof that the formula is not zero works

Powersum formula for differential resolvents

We will prove that we can specialize the indeterminate α in a linear differential α-resolvent of a univariate polynomial over a differential field of characteristic zero to an integer q to obtain a q-resolvent. We use this idea to obtain a formula, known as the powersum formula, for the terms of the α-resolvent. Finally, we use the powersum formula to rediscover Cockle's differential resolvent of a cubic trinomial

A partial factorization of the powersum formula

For any univariate polynomial P whose coefficients lie in an ordinary differential field 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

Differential resolvents of minimal order and weight

We will determine the number of powers of α that appear with nonzero coefficient in an α-power linear differential resolvent of smallest possible order of a univariate polynomial P(t) whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of an α-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockle α-resolvent of a trinomial and finish with a related determinantal formula