The reciprocals of solutions of linear ordinary differential equations (Q1068138)

Let \(F\) be a partial differential field of characteristic zero with the derivative operators \(\delta_ 1,...,\delta_ m\), \(U\) be a universal extension of \(F\), \(\overline F\) be an algebraic closure of \(F\) in \(U\), \({\mathcal P}\) be the ring of Picard-Vessiot elements of \(U\) over \(F\) (for the definitions see [\textit{E. R. Kolchin}, Differential algebra and algebraic groups. New York-London: Academic Press (1973; Zbl 0264.12102)]), and let \(E\) be the group of units of \({\mathcal P}\). If \(u\in U^*\) is an exponential over \(\overline F\) (i.e. \(u^{-1} \text{grad}\, u\in \overline F^ m)\), then \(u\in E\). The authors prove the converse when \(F\) is an ordinary field, that is, \(E\) consists of the exponentials over \(\overline F\) only. This result is obtained by means of techniques which originally used formal power series.