Hey everyone,
   I was talking with Ulrich Thiel (http://www.maths.usyd.edu.au/u/thiel/) 
about his Magma package CHAMP (http://thielul.github.io/CHAMP/) and 
comparing it to our implementation of rational Cherednik algebras. One of 
the reasons that he did not chose Sage was because of the speed of 
polynomial arithmetic in Sage. He took some timings and recorded them here:

http://www.maths.usyd.edu.au/u/thiel/math/benchmarks/

I redid the example on my laptop, which has a similar CPU (i7-4720HQ CPU @ 
2.60GHz), with the latest beta to compare:

sage: R.<x,y,z,t> = ZZ[]
sage: %time f = (1+x+y+z+t)^30
CPU times: user 232 ms, sys: 0 ns, total: 232 ms
Wall time: 241 ms
sage: g = f+1
sage: %time temp = f * g
CPU times: user 16min 34s, sys: 8 ms, total: 16min 34s
Wall time: 16min 34s
sage: 16 * 60 + 34
994

>From Ulrich's timings, it seems like we are still loosing quite a lot in 
converting to/from singular. So that might be something that we can improve 
upon. I was thinking people might be interested in this.

Here's also some slides that does some comparisons on multiplication of 
multivariate polynomials:

http://www.orcca.on.ca/conferences/cca2008/presentations/Monagan_DivMultSparsePolynomials.pdf

Also, Ulrich has told me that he wishes the Sage interface to Magma and 
Maple was not just a pexpect interface, and I was asked by someone else at 
the recent Sage days about a Sage interface to Magma. I am not in a 
position to work on this, lacking a version of both, but there is some 
interest in the community about having a good interface.

Best,
Travis

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