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Gröbner complexity. http://www.singular.unikl.de/forum/viewtopic.php?f=10&t=2585 
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Author:  IliaToli [ Mon Mar 13, 2017 8:21 am ] 
Post subject:  Gröbner complexity. 
I have a question. I am aware that the complexity of Gröbner bases is not very well understood in practice, theoretical results apart. I have this ideal with 21 polynomials, 16 variables. Degrees are: 5, 7, 7, 8, 10, 11, 14, 14, 15, 15, 18, 19, 19, 19, 23, 23, 23, 27, 27, 31, 31. I have 10 nodes, 128 GB of RAM each, available to me. How much time should it take to calculate the Gröbner basis with modStd ? I need only an approximate idea. Gave it 7,000 hours of computer time. Will that be enough? The system is 14.1 MB in size. It comes from quantum chemistry. 
Author:  steenpass [ Mon Mar 13, 2017 3:25 pm ] 
Post subject:  Re: Gröbner complexity. 
You can first try to compute a Gröbner basis in positive characteristic, using a ring definition like Code: ring r = 32003, x(1..16), dp; or something similar. You can also switch on Code: option(prot); see http://www.singular.unikl.de/Manual/latest/sing_307.htm. If this computation doesn't finish, then trying modStd() will be hopeless. If it finishes, then you have good chances that modStd() will finish as well. Given the data you mentioned (21 polynomials in 16 variables of degree up to 31 and size 14.1 MB), the system seems to be quite complicated, but you can still try. 
Author:  Guest [ Thu Mar 16, 2017 2:26 am ] 
Post subject:  Re: Gröbner complexity. 
14.1 MB was the size of the system after interred(). I inserted 12 more relations and 12 new variables, did interred(again), and the size of the system became "only" 422 KB. That gives me hope. I have given 7,000 hours to the calculation, in a computer of 10 nodes, 128 GB RAM each, so I hope for the best. The system comes, as I said, from quantum chemistry. Turned one single differential equation into a system of polynomial equations. 
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