-*- 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: