Standard bases in K

13 years 4 months ago

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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...

Thomas Markwig

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