Below we add a few quick comments on the technical topics discussed in the skype calls:

*) See also slides at bottom of (www.qft.physik.uni-freiburg.de/Activities/AlgGeoSAmp-2017).

*) Most algorithms in Singular to compute Groebner bases can be used to compute Groebener bases of ideals and/or of modules (see various commands in slides)

*) Simplify intermediate results: preliminary solution: insert relation into ideal and add sufficiently many parameters as variables. (The groebner-basis computation works with coefficients in the rational functions over a number of parameters.)
Eventually you have to compute in a quotient ring defined via the command "qring" see example 
*) Floating point: precision can be specified in ring definition, see examples 

*) Cofactor matrix & refined output:

  • Use lift command. In particular with option "slimgb" as in manual page .
  • Useful refined information you can get through sres() function which allows to compute the resolution of the Groebner basis. Takes a groebner basis as input.
  • See also slides (www.qft.physik.uni-freiburg.de/Activities/AlgGeoSAmp-2017)

*) Mathematica interface. This works via file exchange and an optimisation based on MathLink is not presently available.

*) Documentation, forum:

*) tutorials & web interface: https://www.singular.uni-kl.de:8003/

*) parallelised versions are coming based on finite fields methods (this works for ideals, but not modules in the moment): see p27 of the slides

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