My tests were done with the 64 bits icas executable and script files available here. This is a statically linked binary, the dynamically linked icas inside the 64 bit giac debian package should give similar timings. Timings on an 8*Xeon(R) CPU X5482 @ 3.20GHz (July 2010). Source code here (or spkg)

Groebner basis

Frederic Han reports the following timings in comparison with Magma. For large problems without symmetries Giac seems to be faster than Magma.

Multivariate polynomial *

SDMP timings are not present since I don't have a copy of a recent maple version and they would not be fully meaningfull anyway since the immediate data type used by SDMP is 192 bits integer (no allocation for these integers). Therefore comparison is done with the fastest free (as in beer) software I know, Trip, which is in competition with SDMP. Further speed improvements for sparse multiplication in giac would require a thread lockfree malloc, or/and another format for the main giac data type (gen) which would containt natively larger integers, but this would require many changes in the code, and could impact other operations (if for example sizeof(gen) would double, we could have direct integers up to around 120 bits, but most symbolic objects would have a size*2 which would slow operations). As far as I understand, Trip uses a thread lockfree malloc, but this has constraints (idle threads wait for tasks from a master thread, each thread has it's own heap) and it's probably not easy at all to use refcounted objects with local heaps.

Multivariate polynomial GCD

Timings are about the same as Magma 2.15-9 for Fermat GCD test 1 to 4 vars, on Z or Z/nZ (magma is known to be the best in this category). Giac timings are followed by magma timings in parenthesis. For comparion, see also Sage wiki.

Combined test (*,/,factor)

Pearce tests, comparison with maple 14. I don't know how the hardware compare, it seems that factorization is faster with giac.