TITLE: Fast computation of a Groebner basis. What is possible?
QUESTION [18 upvotes]: I need to compute a Groebner basis of 18 polynomials in 19 variables the terms of which have degree at most 3. My aim is to exploit a symmetry in a PDE problem and I am not an expert in algebra or computer algebra. Mathematica is now running for 10 days without answer.  
A similar question description here was asked a couple of years ago but because there was no satisfying answer I raise this question again.
I would appreciate if anyone could tell help me with the following two questions

if there is any hope to compute the Groebner basis? 
if so are there fast at best parallel ready to use programms for computing the Groebner basis?

Any answers, comments and reference are appreciated.

REPLY [2 votes]: I have been trying to answer that question myself for some time and haven't found anything definitive in the literature. I can only point to the paper where M4GB is presented where there are some benchmarks comparing its performance against OpenF4, Magma, and FGb (Faugere algorithm that is used in Maple) for the case of $m$ polynomials of $n$ variables with $m=2n$ and $m=n+1$.
It would be a very nice resource for the community to have an independent benchmark of the different libraries similar to the Mathematical Programming community benchmark service offered by Hans Mittelmann.
Exploring a little more, I found this dicussion in the Google group for the Sage developers where there is a comparison of several codes out there to compute Groebner basis. Keep in mind that the performance depends heavily on your ideal.