Standard bases in K[[t1,…,tm]][x1,…,xn]s

Abstract: In this paper we study standard bases for submodules of respectively of their localisation with respect to a -local monomial ordering. The main step is to prove the existence of a division with remainder generalising and combining the division theorems of Grauert–Hironaka and Mora. Everything else then translates naturally. Setting either or we get standard bases for polynomial rings respectively for power series rings as a special case. We then apply this technique to show that the -initial ideal of an ideal over the Puiseux series field can be read of from a standard basis of its generators. This is an important step in the constructive proof that each point in the tropical variety of such an ideal admits a lifting. [Copyright &y& Elsevier]