2
$\begingroup$

My specific situation is that I need to find the kernel of a very large matrix with entries in the ring of Laurent polynomials in 2 variables over the integers. So basically I have two problems:

1) Understanding theoretically how to do this. After some doing some examples, it certainly seems as if row reduction to get a row echelon/hermite normal form analogue is always possible, but I'm not sure how to justify that this.

Presumably someone someone somewhere knows how to deal with this, as, among possible other things, it would be relevant to finding the Alexander invariant of a link, but I haven't been able to find much that (at least to me) seems helpful. This thesis:

https://www2.eecs.berkeley.edu/Pubs/TechRpts/1995/ERL-95-39.pdf

is perhaps relevant, but the author works with Laurent polynomials over a field, not the integers. Also perhaps relevant is the discussion/sources here:

https://mathoverflow.net/questions/33687/ideals-in-the-ring-of-single-variable-laurent-polynomials-with-integer-coefficie

But that's about all I could find.

2). Getting a computer to do this for me. The matrix I'm working with is probably too large to work with by hand so if you know a program that can deal with this sort of problem that would be extremely helpful.

Thanks!

  • 0
    Row reduction is a generally applicable procedure in the usual way: you need only verify that elementary row operations preserve the rank of a matrix as you do for entries in a field. The only thing to be careful about is that scaling will need to be done by units if you are doing a type-2 operation. (For the scale and add/subtract in type $3$ you can see any multiple is fine) As for a computer method, I'm not sure what's out there for generalized rref, so I'm afraid I cannot be much help there, but I'm reasonably confident such a thing exists. – 2017-02-23
  • 0
    Maybe the [GAP-system](https://www.gap-system.org/) is ok. It can handle Gröbner bases and multivariate polynomial division . Also, maybe the Sympy package of Python could interest you. – 2017-02-23

0 Answers 0