An Algorithm for Converting a Degree Gr

An algorithm for converting a Grr obner basis, G 1 of an arbitrary ideal from one ordering < 1 to a Grr obner basis G 2 with respect to another target ordering < 2 is presented. Its behavior does not depend upon the dimension of the input ideal and it works for any positive or zero dimensional ideal. The target ordering can be any ordering including elimination orderings such as pure lexicographic. The algorithm is similar to the FGLM algorithm for basis conversion of zero dimensional ideals in the following sense: it incrementally constructs (i) a set T of terms, (ii) a set G of polynomials by computing the normal forms of the terms in T with respect to < 1 and solving some linear equations similar to those in FGLM. However the order in which terms are generated in T is not completely determined by the target ordering < 2. If < 2 does not have the property that for every term t, there are only nitely many terms smaller than t, then an enumeration ordering < e is built using < 1 and < 2 such that it has that property. For termination condition, it is rst checked whether all the polynomials in G 1 reduce to 0 modulo G, and if so, then check if G is a Grr obner basis. We prove that there is a T which will be generated within a nite number of steps for which the termination condition would succeed. The algorithm has been implemented in GEOMETER and has been successfully tried on a number of examples. It seems to work better than computing a Grr obner basis directly for ideals with small dimensions. Planned extensions include development of heuristics for developing good enumeration orderings as well as an eecient termination check. For the latter, the use of a Hilbert function of an ideal is being investigated.