Article - post-print
Classically, Grobner 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 k(x, y) that are suitable for this framework. For these valuations, we compute ν(k[x, y] ∗ ) and use this to perform computations concerning ideals in the polynomial ring k[x, y]. Interestingly, for these valuations, some ideals have a finite Grobner basis with respect to the valuation that is not a Grobner basis with respect to any monomial order, whereas other ideals only have Grobner bases that are infinite with respect to the valuation.
This is an author-manuscript of an article accepted for publication in Journal of Symbolic Computation following peer review. The version of record: Edward Mosteig, Value monoids of zero-dimensional valuations of rank 1, Journal of Symbolic Computation, Volume 43, Issue 10, 2008, Pages 688-725, is available online at: http://dx.doi.org/10.1016/j.jsc.2008.01.005.
Edward Mosteig, Value monoids of zero-dimensional valuations of rank 1, Journal of Symbolic Computation, Volume 43, Issue 10, 2008, Pages 688-725, http://dx.doi.org/10.1016/j.jsc.2008.01.005.