Standard bases in K[[t(1),..., t(m)]][x(1),..., x(n)](s)
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. (C) 2008 Elsevier Ltd. All rights reserved.
monomial ordering division with remainder power series standard basis