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Hi!

Sorry, I did not notice the message before.

There exists no 'best variant' for large systems.

The ugly answer is, that you have to try several variants and the most important options, at least redTail... You can also try several orderings. Usually, the best ordering is dp, but sometimes you can arrange the variables in a smart block ordering structure, which makes things much easier.

Maybe, you can also try slimgb, which has had good results in the past for large system, but systems of these number of variables are reallly hard. Having 50000 equations however is awesome and improves the possibility to compute the GB.

The core algorithms std and slimgb both implement their own preprocessing.

Regardings primary decomposition of such big systems. This is more difficult than Groebner bases. So you can essentially forget about it, unless there are some special conditions. In fact, when you're ideal is such over determined and you might only have a few solutions with multiplicity 1, the prime decomposition is equivalent to solving.

I think you did not mention the characteristic.

Cheers, Michael