Irreducible linear differential equations of prime order
Let \(L(y)= \sum_{i=0}^ p a_ i (d^ iy/ dx^ i) =0\) be an irreducible differential equation of prime order where \(a_ i\) lie in a differential field \(k\) with an algebraically closed field of constants. Suppose the differential Galois group of \(L(v) =0\) is unimodular, and \({\mathcal P}_ R= \{P_ 1, P_ 2, \ldots\}\) is the set of all minimal polynomials of an algebraic solution of minimal degree of the associated Riccati equation \(R(u) =0\). In case the differential Galois group \({\mathcal G} (L)\) of \(L(y)=0\) is an imprimitive subgroup of \(\text{SL} (p, \mathbb{C})\), then (1) \({\mathcal P}_ R\) is non empty and consists of polynomials of degree \(p\). (2) If \({\mathcal G} (L)\) has no faithful central monomial representation of degree \(p\), then \({\mathcal P}_ R\) contains at most \((p+1)\) elements and there is a bijection between maximal normal abelian subgroups of \({\mathcal G} (L)\) and elements of \({\mathcal P}_ R\). (3) If \({\mathcal G} (L)\) has no faithful central monomial representation of degree \(p\) and is not isomorphic to a split extension of a cyclic group \(C\) of order dividing \(p-1\) and \(N_ p\) which is the unique (up to isomorphism) non abelian subgroup of order \(p\) in \(\text{SL} (p, \mathbb{C})\), then \({\mathcal P}_ R\) contains exactly one element. If \(L(y) =0\) has a Liouvillian solution over \(k\), then either (or both) of the following holds namely, \({\mathcal P}_ R\) contains exactly one element or \({\mathcal G} (L)\) belongs to a finite set of finite subgroups of \(\text{SL} (p, \mathbb{C})\). In the latter case all solutions of \(L(y) =0\) must be algebraic and we can use the method of the author and \textit{M. Singer} [J. Symb. Comput. 16, 37-73 (1993; Zbl 0802.12005)] to compute the minimal polynomial of a solution of \(L(Y) =0\). The author illustrates the method by working out examples of second and third order equations. Finally, necessary conditions are given for imprimitive finite subgroups which exclude certain finite subgroups of \(\text{SL} (p, \mathbb{C})\) from the list of possible differential Galois groups of a given equation and this extends the results of the author with Singer given for finite primitive groups.
irreducible differential equation of prime order
scientific article; zbMATH DE number 762069