Trac #32686: points_of_bounded_height for projective space is incorrect

The method for number fields iterates over the points of bounded height
for each coordinate. This includes all the appropriate points, but
includes some other points that are larger than the specified bound.

{{{
B=3
K.<v>=QuadraticField(3)
P.<x,y,z>=ProjectiveSpace(K,2)
for Q in P.points_of_bounded_height(bound=B):
    if exp(Q.global_height()) > B+0.001:
        print(Q,exp(Q.global_height()))
}}}

See also #31400

URL: https://trac.sagemath.org/32686
Reported by: bhutz
Ticket author(s): Jing Guo
Reviewer(s): Alexander Galarraga, Ben Hutz

src/doc/en/reference/references/index.rst

@@ -3679,6 +3679,10 @@ REFERENCES:
             and its use for certified homotopy continuation of systems of plane
             algebraic curves, :arxiv:`1505.03432`
 
+.. [Krumm2016] Daid Krumm, *Computing Points of Bounded Height in Projective Space over a Number Field*,
+               MATHEMATICS OF COMPUTATION, Volume 85, Number 297, January 2016, Pages 423–447.
+               http://dx.doi.org/10.1090/mcom/2984
+
 .. [KR2001] \J. Kahane and A. Ryba. *The hexad game*, Electronic
             Journal of Combinatorics, **8**
             (2001). http://www.combinatorics.org/Volume_8/Abstracts/v8i2r11.html

src/sage/dynamics/arithmetic_dynamics/projective_ds.py

@@ -3369,13 +3369,18 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             sage: g = f.affine_preperiodic_model(0, 1); g
             Dynamical System of Projective Space of dimension 2 over Rational Field
               Defn: Defined on coordinates by sending (x : y : z) to
-                    (-x^2 : 2*x^2 + 2*x*y + y^2 : 2*x^2 + 2*x*y + 2*y^2 - 2*y*z + z^2)
+                    (-x^2 : -2*x^2 + 2*x*y - y^2 : 2*x^2 - 2*x*y + 2*y^2 + 2*y*z + z^2)
 
         We can check that ``g`` has affine fixed points::
 
             sage: g.periodic_points(1)
-            [(-1 : 1 : 1), (-1/2 : 1/2 : 1), (-1/2 : 1 : 1), (-1/3 : 2/3 : 1), (0 : 0 : 1),
-            (0 : 1/2 : 1), (0 : 1 : 1)]
+            [(-1 : -1 : 1),
+            (-1/2 : -1 : 1),
+            (-1/2 : -1/2 : 1),
+            (-1/3 : -2/3 : 1),
+            (0 : -1 : 1),
+            (0 : -1/2 : 1),
+            (0 : 0 : 1)]
 
         ::
 
@@ -3384,8 +3389,8 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             sage: f.affine_preperiodic_model(0, 1)
             Dynamical System of Projective Space of dimension 2 over Finite Field in z2 of size 3^2
                   Defn: Defined on coordinates by sending (x : y : z) to
-                        ((z2 + 1)*x^2 : (z2 + 1)*x^2 + (z2 + 1)*x*y + (-z2 - 1)*y^2 :
-                        (z2 - 1)*x^2 + (z2 - 1)*x*y - y^2 + (-z2)*y*z + z^2)
+                        ((-z2)*x^2 : z2*x^2 + (-z2)*x*y + (-z2)*y^2 :
+                        (-z2)*x^2 + z2*x*y + (z2 + 1)*y^2 - y*z + z^2)
 
         ::
 
@@ -3396,9 +3401,8 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             Dynamical System of Projective Space of dimension 2 over
             Univariate Polynomial Ring in c over Finite Field of size 3
               Defn: Defined on coordinates by sending (x : y : z) to
-                    ((2*c^4 + c^3)*x^2 : (2*c^4 + c^3)*x^2 + (2*c^4 + c^3)*x*y + (c^4 + 2*c^3)*y^2 :
-                    c^3*x^2 + c^3*x*y + (2*c^3 + 2*c^2)*y^2 + (c^3 + 2*c^2)*y*z + (2*c^4 + 2*c^3 +
-                    2*c^2)*z^2)
+                    (2*c^3*x^2 : c^3*x^2 + 2*c^3*x*y + 2*c^3*y^2 :
+                    2*c^3*x^2 + c^3*x*y + (c^3 + c^2)*y^2 + 2*c^2*y*z + c^2*z^2)
 
         ::
 
@@ -3409,8 +3413,7 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             Dynamical System of Projective Space of dimension 2
             over Cyclotomic Field of order 3 and degree 2
               Defn: Defined on coordinates by sending (x : y : z) to
-                    (x^2 + y^2 + (-k + 2)*x*z - 2*y*z + (-k + 3)*z^2 :
-                    -2*x^2 + (k - 4)*x*z + (k - 3)*z^2 : -x^2 + (k - 2)*x*z + (k - 2)*z^2)
+                    (-y^2 : x^2 : x^2 + (-k)*x*z + z^2)
 
         ::
 
@@ -3429,8 +3432,7 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             Dynamical System of Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
               2*y - z
               Defn: Defined on coordinates by sending (x : y : z) to
-                    (2*x^2 + y^2 + 4*x*z - 2*y*z + 4*z^2 : -x^2 - y^2 - 2*x*z + 2*y*z - 3*z^2 :
-                    -x^2 - 2*x*z - 2*z^2)
+                    (-x^2 - y^2 : y^2 : x^2 + z^2)
 
         TESTS::