-*- M2 -*-
Title: Noether Normalization
Description:
Implement Noether normalization for quotients of polynomial rings over
fields. This is another good "getting started" project, and one which
would be useful to many people.
The result might be a ring map from a polynomial ring, but Dan should
be consulted about a good interface. Issues to think about: (1) doing
this with a change of variables which is as sparse as possible, (2)
What about characteristic p > 0, in small characteristic?
See:
Eisenbud, Commutative Algebra
alg-geom/9305010 Finding Sparse Systems of Parameters. David Eisenbud, Bernd Sturmfels.
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.
Bermejo, Isabel; Gimenez, Philippe Saturation and Castelnuovo-Mumford
regularity. J. Algebra 303 (2006), no. 2, 592-617. (Reviewer: Yukihide
Takayama) 13D45 (13P10)
There is code in Singular for this too, which might be worth consulting.
Amir Hashemi
``Efficient Computation of Castelnuovo-Mumford Regularity and Satiety''
Submitted to: International Symposium on Symbolic and Algebraic Computation (ISSAC) 2007.
http://www-calfor.lip6.fr/~hashemi/pub.html
Amir Hashemi
``Efficient Algorithms for Computing Noether Normalization''
Submitted to: Special Issue of Mathematics in Computer Science on Symbolic and Numeric Computation.
http://www-calfor.lip6.fr/~hashemi/pub.html
It might be good to incorporate Hashemi's explicit examples of ideals into a
package.
We might need two routines, "noetherPosition" and "noetherNormalization".
Mike says: see the papers of Hashemi and Bermejo above. A generic change of
coordinates gives a new Borel-fixed ideal with the same regularity. The papers
give a way to test whether a specific change of coordinates is general enough
to have the property, in which case we say the ideal has been moved to strong
Noether position. See the Singular package called "m-regular" or something
like that. We could write code to do this, too. This could be a new new
function called "strongNoetherPosition" and a new option to "regularity" that
would call upon it.
Mike also says: it might be useful to have the version of noetherNormalization
that normalizes a chain of ideals.
=============================================================================
Proposed by: Mike Stillman
Advisor: Mike Stillman
Project assigned to: Bart Snapp ,
Nathaniel Stapleton
Current status: complete, with the following idea for future work:
The current implementation seems to work. One might be able to increase
efficiency if instead of using the lexicographical ordering, a block
ordering is used instead.
=============================================================================
Progress log:
We found two very helpful resources:
Brumatti, P.; Simis, A.; Vasconcelos, W. V. Normal Rees algebras. J. Algebra
112 (1988), no. 1, 26-48.
Logar, Alessandro(I-TRST) A computational proof of the Noether normalization
lemma. Applied algebra, algebraic algorithms and error-correcting codes (Rome,
1988), 259-273, Lecture Notes in Comput. Sci., 357, Springer, Berlin, 1989.