Dear Thierry, I have to solve polynomial systems of equations on a regular basis and would be happy to see sage improve in this respect.
My strategy is to use numerical algorithms. There is Bertini (I glued together my own customized interface to it 4-5 years ago). They now migrated to C code and are available on github: https://github.com/bertiniteam/b2 Bertini can guarantee "with probability 1" that it found _all_ isolated solutions (under the right circumstances). Then, using some degree bounds and solving the system around an isolated solution to get enough decimal places, it is possible to get back the actually minimal polynomial for the algebraic solution. This method was successfully used for example in: https://page.mi.fu-berlin.de/moritz/papers/t2-diss.html https://page.mi.fu-berlin.de/moritz/papers/j002-computing-maximal-copies.html Reading about Bertini, I learned about PHCpack, which is apparently partially available in Sage: http://doc.sagemath.org/html/en/reference/interfaces/sage/interfaces/phc.html http://homepages.math.uic.edu/~jan/download.html http://homepages.math.uic.edu/~jan/PHCpack/phcpack.html There is also a recent development in julia, developed in Berlin, where one of the long term goal is to improve polyhedral homotopies for such systems. Apparently, they can beat bertini in instances of 0-dimensional cases. https://www.juliahomotopycontinuation.org/ That's my bit of knowledge in this area. I know that it is 'numerical' but, as far as applications go, they often help a lot and it is a very active field of research and algorithmic development... Best, JP -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
