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

In this paper we study standard bases for submodules of K[[t1,…,tm]][x1,…,xn]s 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 m=0 or n=0 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 t-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.