-*- 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.