Standard Bases in K[[t_1,...,t_m]][x_1,...,x_n]^s

TL;DR

This paper develops a theory of standard bases for submodules over power series and polynomial rings, generalizing division algorithms and applying to tropical geometry and Puiseux series.

Contribution

It introduces a division with remainder for submodules over power series rings, extending Grauert and Mora's theorems, and applies this to tropical geometry.

Findings

Existence of a division algorithm generalizing Grauert and Mora

Standard bases can be used to determine t-initial ideals in Puiseux series

Facilitates lifting points in tropical varieties

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 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.