Homogeneous \(p\)-differential polynomials. (Q1414671)

The first author earlier showed that algebraic geometry may be enlarged by adjoining a ``Fermat quotient type operation'' and the resulting geometry was named as \(\delta\)-geometry. Later he with his kolleagues introduced certain graded rings that should be interpreted as rings of sections of appropriate line bundles on the corresponding orbit spaces in \(\delta\)-geometry. In some cases these rings are isomorphic to certain subrings \(R\{x\}^{\ast}\) and \(R\{x\}_{\ast}\) of the ring \(R\{x\}\), which is an analogue of the Ritt-Kolchin ring of differential polynomials (cf. \textit{A. Buium} [Vostokov, S. (ed.) et al., Algebraic number theory and algebraic geometry. Papers dedicated to A. N. Parshin on the occasion of his sixtieth birthday. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 300, 59--97 (2002; Zbl 1084.14524)] for an overview). In the present paper the authors study the properties of this ring which is ``\(\delta\)-geometric''. In particular, the \(p\)-adic completion of a basic ring of fractions attached to \(R\{x\}_{\ast}\) is described and the analysis of homomorphisms between the basic rings is provided.