Standard bases in K

In this paper we study standard bases for submodules of K[[t1, . . . , tm]][x1, . . . , xn]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. The paper follows the lines of [GrP02] and [DeS07] generalising the results where necessary. Basically, the only original parts for the standard bases are the proofs of Theorem 2.1 and Theorem 3.3, but even here they are easy gene...

• Type: Journal
• Year: 2008
• Where: JSC
• Authors: Thomas Markwig
• Keywords: Ideal, JSC 2008, Power Series, Standard Bases