# Memory usage of Gröbner basis computation

I've been calculating some Gröbner bases in preparation for finding non-commutative Hilbert series (and, once I recreate that, characters of group actions). Specifically, I've been using the letterplace implementation, which I believe calculates to degree $d$ the Gröbner basis for an ideal in the non-commutative free algebra with $n$ generators by passing to something in a commutative ring with $nd$ generators.

In the two cases I've worked with, I can get to degree $5$ with a $9$-generator algebra or to degree $4$ with a $12$-generator algebra; doing one degree higher has led to exceeding the 24GB (8GB RAM + 16GB swap) of the Linux machine with the most memory I can access. This suggests that I can get somewhere in the 40s in terms of number of variables but not the 50s. How hopeless is the growth in memory access for this algorithm; would trying to jerry-rig in a different algorithm for the commutative algebra part be worthwhile?

• For this sort of things, I think it is better to work experimentally ("profile, don't speculate" as the motto goes): try to find toy versions of your problem with different complexities, and see how the memory taken (or total final number of terms) increases with the parameters. (I also use this approach to try to find the most efficient term order, when there is choice and it's not obvious beforehand.) Sep 19, 2017 at 7:21

You should also consider what field to compute in. Could you get lots of useful information (approximation?) out of a GB computed over $\mathbb{Z}_p$ for example?