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arXiv · 0705.2436

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

Thomas Markwig
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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.

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BibTeXRIS

Thomas Markwig. 2007-05-16. Standard Bases in K[[t_1,...,t_m]][x_1,...,x_n]^s. https://arxiv.org/abs/0705.2436

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