Accessing additional MSolve functionality?

[antonydellavecchia]

The current version of rational_solutions uses a lex ordering and does a groebner basis computation through singular by default without the option to change either.

The question is if or how we should access additional functionality in AlgebraicSolving to speed up the computation, in particular for rational coefficients.

This has been discussed awhile ago, now we reopen the discussion concerning e.g. aspects of randomization.

Probably we should also add the real_solutions to documentation on the same page as rational_solutions.
In fact it could be nice to have a general page on polynomial system solving for beginners etc.

@thofma @fieker @ederc @lkastner @micjoswig @mohabsafey

[ederc]

The current version of rational_solutions uses a lex ordering and does a groebner basis computation through singular by default without the option to change either.

AFAIK this is the fallback version. There is rational_solutions() specific for ideals over QQ which calls _rational_solutions() which is the call to msolve.

Maybe @thofma knows more about the details.

[ederc]

Probably we should also add the real_solutions to documentation on the same page as rational_solutions.
In fact it could be nice to have a general page on polynomial system solving for beginners etc.

There is already a tutorial on this topic.

[micjoswig]

Probably we should also add the real_solutions to documentation on the same page as rational_solutions.
In fact it could be nice to have a general page on polynomial system solving for beginners etc.

There is already a tutorial on this topic.

Thanks for pointing this out. This is useful, indeed.

I was confused by the current documentation. We have

https://docs.oscar-system.org/dev/CommutativeAlgebra/ideals/#Vanishing-sets-and-solving-multivariate-polynomial-systems

but this only mentions rational_solutions.

Since solving polynomial equations is such an important topic, should we move that fragment of the docs one level higher up (directly beneath Commutative Algebra)? There we could also list real_solutions and a link to the tutorial.

Good idea?

[antonydellavecchia]

The current version of rational_solutions uses a lex ordering and does a groebner basis computation through singular by default without the option to change either.

AFAIK this is the fallback version. There is rational_solutions() specific for ideals over QQ which calls _rational_solutions() which is the call to msolve.

Maybe @thofma knows more about the details.

Ah sorry, we missed that yesterday when looking at the code. So on line 204 there is method implementation

function rational_solutions(K::Field, I::MPolyIdeal)
  R = base_ring(I)
  _, f = change_base_ring(K, R)
  return rational_solutions(f(I))
end

function rational_solutions(I::MPolyIdeal{QQMPolyRingElem})
  # Call msolve/AlgebraicSolving
  return _rational_solutions(I)
end

Which uses MSolve when the ideals are over QQ

[jankoboehm]

Is this then verified? If not, we should make sure that this information arrives at the user, see #5499, and should the standard algorithmic option then be a verified one? Also note that the current fallback implementation is very restricted, the fix will be #5495.