On Mon, 7 Jan 2019 at 09:51, jplab <jeanphilippela...@gmail.com> wrote:
> 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. > another approach is via sum of squares (sos) relaxations (Lasserre’s). So an essential ingredient is an SDP solver, and this is already in Sage (the only backend is not a very fast one though) I also have some draft code for sos building of SDPs. > 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 > great, it seems they finally went properly open-source! Would be good to hook it up in Sage... Dima > <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. > -- 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.
