Quillen Suslin_(IMA2011)

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title: Quillen-Suslin (IMA2011) permalink: wiki/Quillen-Suslin_(IMA2011)/ layout: wiki

The Quillen-Suslin Theorem states that any projective module M over a polynomial ring R[x_1,...,x_n], with R a PID, is free. Given a presentation of a projective module M by generators and relations, we can algorithmically construct a free generating set for M.

QuillenSuslin Package for Macaulay2

• Current status of the package:
  • Work began during the 2010 Colorado College Macaulay2 workshop and has continued over the past year.
  • All algorithms implementing a modification of the Logar-Sturmfels algorithm for computing a free basis of a projective module over R[x_1,...,x_n] with R = QQ, ZZ, or ZZ/p with p a prime integer have been completed. A preliminary report on the package is in progress.

• Potential areas for expansion:
  • Use the existing code or slight modifications thereof to implement Suslin's Stability Theorem, which factors a matrix in SL_n(k[x_1,...,x_m]) into a product of elementary matrices for n \geq 3 and m \geq 1. A paper by H. Park and C. Woodburn detailing an algorithm for such a factorization is available at http://www.math.uiuc.edu/K-theory/0019/.
  • Extend the algorithm to work over Laurent polynomial rings. There is some information about algorithms in the following two papers:

H. Park, Symbolic Computation and Signal Processing (2004) (See "Park's Causal Conversion Algorithm", pg. 218. This algorithm is 'nice' in the sense that it reduces the problem to completing a unimodular row over a polynomial ring, for which all of the code has been written. The downside to this is that the exponents on the elements of the matrix grow exponentially.)

M. Amidou & I. Yengui, An Algorithm for Unimodular Completion over Laurent Polynomial Rings (2008)

Useful Papers/Links

• Logar, Alessandro; Sturmfels, Bernd, Algorithms for the Quillen-Suslin theorem. J. Algebra 145 (1992), no. 1, 231--239.
• Laubenbacher, Reinhard C.; Woodburn, Cynthia J. A new algorithm for the Quillen-Suslin theorem. Beiträge Algebra Geom. 41 (2000), no. 1, 23--31.

• An algorithm to compute such a generating set for a projective module has already been implemented in Maple, and the documentation for the Maple package can be found at:

QuillenSuslin Maple Package Documentation

• The author of this package also has some notes about the relevant algorithms, which can be found on pages 25-43 of the following PhD thesis:

http://wwwb.math.rwth-aachen.de/~fabianska/PSLHomepage/DissertationAF.pdf

To-Do List

• (Done) Implement Park's 'causal conversion algorithm' to return a product of elementary matrices consisting of Laurent polynomials as well as an invertible change of variables which convert a unimodular row of Laurent polynomials into a unimodular row of polynomials.
• (Done) Implement Laurent polynomial versions of the core methods of the QuillenSuslin package, such as qsAlgorithm, completeMatrix, and getFreeBasis.
• Implement the strategy for unimodular completion of a row of Laurent polynomials given by Amidou-Yengui in the above paper. Make this an optional strategy of the methods mentioned above, via Strategy => Park or Strategy => AY.
• Merge methods for Laurent polynomial rings into current QuillenSuslin package and have the methods automatically determine the appropriate algorithm based on whether the matrix or module is defined over a QuotientField or a PolynomialRing.

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