Standard bases in [formula omitted

Abstract

In this paper we study standard bases for submodules of K [ [ t 1 , … , t m ] ] [ x 1 , … , x n ] s respectively of their localisation with respect to a t ¯ -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.