2005/04/13 by Edward Mosteig, Mosteig, Edward
Mathematics · #12J20 #12Y05 #13P10 #68W30 #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #msc:12J20 #msc:12Y05 #msc:13P10 #msc:68W30
paper · pdf · doi:10.48550/arxiv.math/0504263
23 pages
arxiv created 2005/04/13 · openalex publication_date 2005/04/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build on these ideas by providing a class of valuations on rational function fields of two variables that are suitable for this framework. For these valuations, we explicitly compute the image of an underlying polynomial ring and use this to perform computations concerning ideals in the polynomial ring. Interestingly, for these valuations, some ideals have a finite Groebner basis with respect to the valuation that is not a Groebner basis with respect to any monomial order, whereas other ideals only have Groebner bases that are infinite with respect to the valuation.