-*- M2 -*- Title: Generic Initial Ideals Description: See Eisenbud's book on Commutative Algebra for the definition, and for many of the very useful properties of generic initial ideals. See also: Green, Mark; Stillman, Michael A tutorial on generic initial ideals. Gröbner bases and applications (Linz, 1998), 90--108, London Math. Soc. Lecture Note Ser., 251, Cambridge Univ. Press, Cambridge, 1998. Bermejo, Isabel; Gimenez, Philippe; Morales, Marcel Castelnuovo-Mumford regularity of projective monomial varieties of codimension two. J. Symbolic Comput. 41 (2006), no. 10, 1105-1124. This is a good "getting started" project. One way to probabilisitically compute the generic initial ideal of an ideal (with respect to a term order) is to make a random change of coordinates, and take the lead term ideal of the resulting ideal. This is only probabilistic: it could give the wrong answer. One way around this is to do the above several times, and take a consensus. The choice of how many times to do it should probably be an optional argument. An interesting research project would be to find a method to certify that one has found the generic initial ideal. One way is to add n^2 variables (if the original polynomial ring has n variables), and to make the "generic" change of coordinates. Unfortunately, this is often computationally prohibitive. There is a rough template for this package already, at M2/Macaulay2/packages/GenericInitialIdeals.m2, but it has not been written yet. Many people would like to see this in M2! -- remarks from David Eisenbud: Giulio Caviglia had some interesting ideas on implementing generic initial ideals (or was it initial ideals with the same useful properties?) by "generalizing" one variable at a time, and using a test for a given one to be general enough -- I'm afraid I don't now remember the details. Giulio, please fill us in! If I give you an ideal and assert that the (revlex, say) initial ideal in the given coords is generic, is there any test to say whether this is so? This would be a good research project, if not! A related set of ideas is in an old paper of mine with Bernd Sturmfels: we were looking for sparse Noether Normalizations -- very similar to looking for sparse changes of coordinates that would lead to generic, or sufficiently generic, initial ideals. The reference, for what it's worth, is MR1282836 (95i:13020) Eisenbud, David ; Sturmfels, Bernd . Finding sparse systems of parameters. J. Pure Appl. Algebra 94 (1994), no. 2, 143-15. ============================================================================= Proposed by: Mike Stillman Potential Advisor: Mike Stillman Project assigned to: Nathaniel Stapleton and Alexandra Seceleanu Current status: done: code has been written, the package GenericInitialIdeal is done, is part of the M2 source code. ============================================================================= Progress log: