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TitleStandard bases in K[[t 1,…, t m]][x 1,…,x n] s
Author(s) Thomas Markwig
TypeArticle in Journal
AbstractIn 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.
KeywordsStandard basis, Monomial ordering, Division with remainder, Power series
URL http://www.sciencedirect.com/science/article/pii/S0747717108000461
JournalJournal of Symbolic Computation
Pages765 - 786
Translation No
Refereed No