Standard bases in mixed power series and polynomial rings over rings

In this paper we study standard bases for submodules of a mixed power series and polynomial ring RJt1, . . . , tmK[x1, . . . , xn] s respectively of their localization with respect to a t-local monomial ordering for a certain class of noetherian rings R. The main steps are to prove the existence of a division with remainder generalizing and combining the division theorems of Grauert–Hironaka and Mora and to generalize the Buchberger criterion. Everything else then translates naturally. Setting eitherm = 0 or n = 0 we get standard bases for polynomial rings respectively for power series rings over R as a special case. The paper follows to a large part the lines of [Mar10], or alternatively [GrP02] and [DeS07], adapting to the situation that the coefficient domain R is no field. We generalize the Division Theorem of Grauert–Hironaka respectively Mora (the latter in the form stated and proved first by Greuel and Pfister, see [GGM94], [GrP96]; see also [Mor82], [Grä94]). The paper should therefore be seen as a unified approach for the existence of standard bases in polynomial and power series rings for coefficient domains which are not fields. Standard bases of ideals in such rings come up naturally when computing Gröbner fans (see [MaR15a]) and tropical varieties (see [MaR15b]) over non-archimedian valued fields, even though we consider a wider class of base rings than actually needed for this. An important point is that if the input data is polynomial in both t and x then we can actually compute the standard basis in finite time since a standard basis computed in R[t1, . . . , tm]〈t1,...,tm〉[x1, . . . , xn] will do. Many authors contributed to the further development (see e.g. [Bec90] for a standard basis criterion in the power series ring) and to generalizations of the theory, e.g. to algebraic power series (see e.g. [Hir77], [AMR77], [ACH05]) or to differential operators (see e.g. [GaH05]). This list is by no means complete. Date: March, 2015. 1991 Mathematics Subject Classification. Primary 13P10, 13F25, 16W60; Secondary 12J25, 16W60.