A point on a Riemann surface belongs to type RieSrfPt. At this point it is necessary to describe the representation of such points
A number of ways of creating points on a Riemann surface will be described. Some functions returning points have already been met: BasePoint(X), RamificationPoints(X) and InfinitePoints(X). The most important way of creating a point on a Riemann surface is to coerce a sequence/tuple into the surface.
Attempt to create a point on the Riemann surface X from the sequence or tuple S. In order to succeed S has to be a sequence or tuple of length 2 (resp. 3) that is coercible into a complex field and satisfies the affine (resp. projective) defining equation for X. A special case is the points at infinity of a superelliptic Riemann surfaces which may be defined by [k] where k is an integer. If successful, the intrinsic returns true and the point on X. Otherwise false is returned.
Attempt to create a point on the Riemann surface X from the sequence S by calling IsCoercible(X,S). In order to succeed, S has to be a sequence of length 2 (resp. 3) that is coercible into a complex field and satisfies the affine (resp. projective) defining equation for X. A special case is that of the points at infinity of superelliptic Riemann surfaces which may be defined by [k] where k is an integer.
If X is a non-superelliptic Riemann surface and S is a tuple defining a possible point of X, the intrinsic IsCoercible(X,S is called to test if S defines a valid point and if so, produce the point of X corresponding to S. The tuple S has to be of the form <x, s> where x is either coercible into a complex field or Infinity() and s ∈{1, .., m}. If successful the intrinsic returns true, and the point on X. Otherwise false is returned.
> Qxy<x,y> := PolynomialRing(Rationals(),2); > f := -x^7 + 2*x^3*y + y^3; > X := RiemannSurface(f); > X; Riemann surface of genus 2 defined by: 0 = -x^7 + 2*x^3*y + y^3 and prescribed precision 30The point of X defined by x = [ - 2.000000000, - 6.085518827] and s = 2 will be defined.
> b, pt := IsPoint(X, < [-2.000000000, -6.085518827], 2 >); > b; true > pt; (-2.000000000 - 6.085518827*I, -23.68417228 + 74.84801836*I)
Given a point P belonging to the Riemann surface X, return X.
Given a point P belonging to a Riemann surface X a representation of P is returned. If P can be uniquely represented by a pair (x, y) ∈C2, a sequence containing these numbers is returned. In other cases, such as for infinite points or singular points of the underlying affine (projective) curve, the representation is a 2-tuple < x, S > where x ∈C ∪{ ∞} and S ⊂{1, .., m} is an indexed subset. The only special case is for points at infinity for superelliptic Riemann surfaces, which are represented by < ∞, k > where k in an integer.
Given a point P belonging to a Riemann surface X, the coordinates of P in the projective closure of the affine curve are returned. Note that these coordinates are not necessarily unique, as they may correspond to singular points.
The ramification index of the point P belonging to a Riemann surface is returned.
Given a Riemann surface X and an integer k in the range [0 ... t], where t is the number of discriminant points for X, the points lying over the k-th discriminant point are returned. If k is zero, the points lying over infinity are returned.
Ht: RngIntElt Default: 10^5
Finite: BoolElt Default: true
A randomly generated point on the Riemann surface X is returned. If the parameter Finite is set to false a random infinite point is returned. The second parameter Ht, allows the user to bound the absolute value of the x-coordinate.
> Qxy<x,y> := PolynomialRing(Rationals(),2); > f := -4*x^4 - 5*x^3*y + x^3 + 2*x^2*y^2 - 5*x^2*y + 3*x^2 > + 3*x*y^3 + x*y - 5*x - 8*y^3 - 3; > X := RiemannSurface(f);
An easy way of defining a point on a Riemann surface X is via a pair of coordinates, using the fiber attribute of X.
> L := X`Fiber(0);
> Pts := [ X![0,L[i]]: i in [1..#L] ];
> Pts;
[
(0.0000000000, -0.7211247851),
(0.0000000000, 0.3605623926 - 0.6245123832*I),
(0.0000000000, 0.3605623926 + 0.6245123832*I)
]
Using the Riemann surface Print command it is easy to control the precision used when printing objects. The various ways of printing the infinite points will also be demonstrated.
> Print(Pts[1] : Precision := Precision(X)); (0.000000000000000000000000000000, -0.721124785153704191160819155390) > Representation(Pts[1]); [ 0.000000000000000000000000000000000000000000000000000000, -0.721124785153704191160819155390054794195934626749675288 ]
Next the various ways of printing the infinite points are shown.
> InfinitePoints(X);
[
Point at infinity on sheets {@ 1 @},
Point at infinity on sheets {@ 2, 3 @},
Y-infinite point over x = 2.666666667 on sheets {@ 2 @}
]
> < Representation(P): P in InfinitePoints(X) >;
<<Infinity, {@ 1 @}>, <Infinity, {@ 2, 3 @}>,
<2.66666666666666666666666666666666666666666666666666667, {@ 2 @}>>
> X!<Infinity(),1> eq InfinitePoints(X)[1];
true
> X!<Infinity(),2> eq X!<Infinity(),3>;
true
> X!<8/3,2> eq InfinitePoints(X)[3];
true
Using the intrinsic Coordinates, the projective coordinates of the points can be retrieved.
> [ Coordinates(Pt): Pt in InfinitePoints(X) ];
[
[ 0.750000000000000000000000000000, 1.00000000000000000000000000000,
0.000000000000000000000000000000 ],
[ -1.00000000000000000000000000000, 1.00000000000000000000000000000,
0.000000000000000000000000000000 ],
[ 0.000000000000000000000000000000, 1.00000000000000000000000000000,
0.000000000000000000000000000000 ]
]
> X![3/4,1,0] eq InfinitePoints(X)[1];
true
> X![-1,1,0] eq InfinitePoints(X)[2];
true
> X![0,1,0] eq InfinitePoints(X)[3];
true
Next the ramification points will be printed.
> RamificationPoints(X);
[
(-0.9945334268 - 0.01315512874*I, 0.05451089484 - 0.1643916971*I),
(-0.9945334268 + 0.01315512874*I, 0.05451089484 + 0.1643916971*I),
(-0.5252377944 + 0.03223401255*I, 0.01450803309 - 0.2043107259*I),
(-0.5252377944 - 0.03223401255*I, 0.01450803309 + 0.2043107259*I),
(0.8365927809 + 0.5016034840*I, -0.4936743728 + 0.5748084711*I),
(0.8365927809 - 0.5016034840*I, -0.4936743728 - 0.5748084711*I),
(0.6518240243 - 0.8906693804*I, -1.581225731 - 0.3052027833*I),
(2.549953730, 19.14988442),
(0.6518240243 + 0.8906693804*I, -1.581225731 + 0.3052027833*I),
Point at infinity on sheets {@ 2, 3 @}
]
The discriminant points and the points lying over them will be examined.
> DiscriminantPoints(X)[1];
-0.994533426780725243594285430344169336513181415875424160 -
0.0131551287359799949411725445257194879581749598493457862*I
> PointsOverDiscriminantPoint(X,1);
[
(-0.9945334268 - 0.01315512874*I, 0.05451089484 - 0.1643916971*I),
(-0.9945334268 - 0.01315512874*I, 0.07106577813 + 0.3329009545*I)
]
Points lying over Infinity() are obtained as follows:
> PointsOverDiscriminantPoint(X,0);
[
Point at infinity on sheets {@ 1 @},
Point at infinity on sheets {@ 2, 3 @}
]
Random finite or infinite points on X can be obtained as follows:
> P1 := RandomPoint(X: Finite);
> P1;
(-2.285969242 + 16.06877365*I, -1.575257283 + 21.29854014*I)
> P2 := RandomPoint(X: Finite := false);
> P2;
Y-infinite point over x = 2.666666667 on sheets {@ 2 @}