Re: [sage-devel] Re: Real algebraic varieties

[Dima Pasechnik]

On Mon, 7 Jan 2019 at 09:51, jplab  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

> 
>
> 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
>
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[sage-devel] Re: Real algebraic varieties

[jplab]

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

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