Rational Point algorithm bug

[nkaplan]

When you have a curve defined over a projective space over a finite field, it looks like sometimes the rational point list coming from the Brill-Noether package in Singular is not the full list of rational points.

sage: S.<x,y,z> = GF(5)[]
sage: g = x*z+z^2
sage: G = Curve(g)
sage: G.rational_points('enum')
[(0 : 1 : 0), (1 : 0 : 0), (1 : 1 : 0), (2 : 1 : 0),
 (3 : 1 : 0), (4 : 0 : 1), (4 : 1 : 0), (4 : 1 : 1),
 (4 : 2 : 1), (4 : 3 : 1), (4 : 4 : 1)]
sage: G.rational_points('bn')
[(0 : 1 : 0), (4 : 0 : 1), (4 : 1 : 1), (4 : 2 : 1),
 (4 : 3 : 1), (4 : 4 : 1)]

The problem was how Sage calls Singular's functions.

[minz]

Part of the problem was the same as in the computation of Riemann-Roch spaces ​#8997.

There still remains a problem: The above example works fine now if we replace z by y (which shouldn't make a difference in terms of correctness of the algorithms involved). Maybe the current singular interface makes certain assumptions about the variable names and their ordering that we still need to pay attention to here?

[minz]

correctly wraps Singular's Brill-Noether method and adds doctest

[minz]

We were missing so far that and how Singular dehomogenizes the defining polynomial of the curve: Always with respect to the last variable. If the corresponding line is a component of the curve, the output was missing those points.

[cremona]

The patch looks good to me and applies cleanly to 4.8.alpha4. All long tests in sage/scheme pass. I am happy with this.