Reverse Lexicographic Initial Ideals of Generic Ideals Are Nitely Generated

This article generalizes the well-known notion of generic forms to the algebra R 0 , introduced in 27]. For the total degree, then reverse lexicographic order, we prove that the initial ideal of an ideal generated by nitely many generic forms (in countably innnitely many variables) is nitely generated. This contrasts to the lexicographic order, for which initial ideals of generic ideals in general are non-nitely generated. We use the approximation methods developed in 28], together with the results of Moreno in 18] on \ordinary" initial ideals of reverse lexicographic initial ideals of generic ideals, to prove that a minimal generating set of the initial ideal of an ideal generated by k generic forms is contained in the semi-group M k ; hence, it is nite. As a generalization of this result, we prove that what we call \pure generic" ideals in an non-noetherian overring of a polynomial ring on two groups of variables, have initial ideals (with respect to a \twisted" product order of degrevlex on the two groups) that are nitely generated. The natural question, \is the reverse lexicographic initial ideal of an homogeneous, nitely generated ideal in R 0 nitely generated" is posed, but not answered; we do, however, point out one direction of investigation that might provide the answer: namely to view such an ideal as the \specialization" of a generic ideal.