CoCoA » CoCoA-5

The point is that RationalSolve tries to be clever about deciding whether it should look for solutions in affine space or in projective space. However, it does not tell the user which type of solution it returns.

If all the polys are homogeneous then RationalSolve returns "projective" solutions; otherwise it returns "affine" solutions.

With the args [x,x*y] the solution found is projective, and the answer [0,1] is correct; other apparent solutions such as [0,2] are the same projective point.

With the args [x,x*y,y] the cheap trick used for detecting the special case of an ideal whose radical is the "irrelevant ideal" does not work; so this is regarded as homogeneous input, and the correct answer is that there are no rational projective solutions.

SUGGESTION it would be better if the function returned an indication of whether the solutions produced are affine or projective.

What should RationalSolve do if the input is homogeneous but the grading is non-standard?

Assuming we still want to have the automatic selection between affine and projective solving, I suggest the following criterion:
if the ideal (generated by the given polys) is zero-dim in the subring generated by the indets actually appearing in the list of polys then use affine solving otherwise if the ideal is homog then use projective solving. If neither case applies then error.

Another example...

use ZZ/(3)[x,y,z];
RationalSolve([x,x*(y-z)]);
--> ERROR: Value must be non-zero
--> [CoCoALib] factor(x)
--> WHERE: at line 102 (column 23) of RationalPoints.cpkg5
-->   LinearFacs := [g in factor(f).factors | deg(g) = 1];
-->                       ^^^^^^^^^

Bernhard Andraschko reportes the following:

Use QQ[x,y];
RationalSolve([x^2+y^2]);
RationalSolve([x^2,y^2]);

I now think it might be better to avoid the "clever" function RationalSolve which tries to guess whether it should look for affine solutions or projective ones.
The heuristic it uses is not entirely transparent, and makes some strange mistakes at times.

Comments? Opinions?

I am wondering about changing the return value of RationalSolve into a record, e.g.
currently we get

/**/ RationalSolve([x^2+y^2-2,x*y-1]);
[[-1,  -1],  [1,  1]]

/**/ RationalSolve([x^2+y^2-2,x*y-1]);
record[indets := [x,  y],  solns := [[-1,  -1],  [1,  1]]]

Comments? Opinions?

Anna has approved the suggestion in comment 13 (just above) about returning a record.
Now I must impl, and revise the doc. Hope to do this soon.

Now I wonder what the names of the fields in the records should be.
It may be sensible to use different names for the list of affine solns, and the list of projective solns.
Or is that against KISS?

Possible names are AffinePts and ProjectivePts (or ProjPts).

PS: I have implemented the suggestion in comment 13, incl revised doc.

Possible names are AffinePts and ProjectivePts (or ProjPts).

I vote for AffinePts and ProjectivePts