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This is a very general/open-ended question.

I wonder if numerical solutions to large scale system of polynomial equations (in many variables) are used in physics at all. I am looking for some good reference to research level (open) problems in physics that involves large scale polynomial systems that cannot be solved by software packages like Mathematica or Maple within any reasonable amount of time.

Ideally, these problems

  • focus on the complex solutions (as oppose to integer or rational solutions)
  • relatively easy to understand for outsiders (with sufficient math background)
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Are you talking about something like Finite Element Analysis? That's the solution of a system of potentially millions/billions of basis functions (typically polynomials) to understand whatever PDE you are interested in (fluid flow, structural mechanics, etc). – tpg2114 Feb 2 at 0:24
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Of course, anything really can be solved by Mathematica and Maple. Perhaps not directly but those are completely capable of numerical/indirect solutions to problems. Given enough time/CPUs of course. – tpg2114 Feb 2 at 0:26
Mathematica/Maple are just unable to solve many polynomial systems. But good point though. I have just edited. It is now "...cannot be solved by Mathematical/Maple within any reasonable amount of time". – ssquidd Feb 2 at 0:40
Anything capable of a for (or do) loop and basic math operations can solve polynomial systems. Which they both have. And both Mathematica and Maple can be used in parallel so it's possible they could solve anything other software could, although I know nothing about how well they scale in parallel. – tpg2114 Feb 2 at 0:43
I think finite element methods generally produce large scale linear systems (at least people prefer that). That's not really what I am looking for. But if there are PDEs that force people to consider (nonlinear) polynomial systems, then they are good problems for me. – ssquidd Feb 2 at 0:44
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closed as not constructive by Chris White, David Zaslavsky♦ Feb 2 at 0:48

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