Polynomial division

[bluescarni]

Some initial code for polynomial division and GCD. Work in progress.

[bluescarni]

@certik @isuruf @rwst pinging in case you are interested

So this is Piranha's first foray into poly division and GCD. I implemented both exact multivariate polynomial division and GCD over generic UFDs.

Just to make things clear since the beginning, performance is going to be horrid. This is due to two root causes:

• as you know, Piranha uses hash tables for representing polynomials. This works well for ring operations, but division and GCD essentially require the terms to be kept in an ordered fashion. I will need to code an alternative data structure into which Piranha's hash tables will have to be converted before entering division and GCD algorithms. I have a fairly precise idea of what to do, but as you can imagine it's rather substantial work and I'd prefer to have something working now rather than delaying merging this PR (which I need in order to implement rational functions);
• this is the first time that I touch division and GCD operations, so I went for easy algorithms to code and implemented them in a straightforward fashion. Currently, there are two GCD algorithms available: subresultant sequence with pseudo-remainders and heuristic GCD. The heuristic GCD is tried first, if it fails the other algorithm will be used. You can also force the algorithm to be used with a flag. There are other families of algorithms which are theoretically better (at least in some cases), so eventually it makes sense to implement at least some of them.

I consider the first point to have a higher priority than the second. If anything, I think we need to have a good implementation of the primitive data structures before comparing algorithmic performance.

Despite the performance problems, the code is as tested as any new feature I have implemented in Piranha so far and at this point I am not aware of correctness bugs. I made sure to have 2 GCD implementations available so I could compare the results of the two as an additional check.

I'll merge the PR tomorrow when all the checks are run.

[certik]

@bluescarni cool, that's exciting. Thanks for letting us know.