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30294: beginning work on step 4.b.
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@EnderWannabe
EnderWannabe committed Aug 5, 2020
1 parent 508921e commit c21fb94
Showing 1 changed file with 97 additions and 94 deletions.
191 changes: 97 additions & 94 deletions src/sage/dynamics/arithmetic_dynamics/berkovich_ds.py
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Original file line number Diff line number Diff line change
Expand Up @@ -878,6 +878,8 @@ def min_res_locus(self, ret_conjugation=False):
r"""
Returns the minimal resultant locus and the minimal order of the resultant.
Only implemented for Berkovich spaces backed by number fields.
Let the action of this map `\phi` be given by
`[X : Y] \to [F(X,Y) : G(X,Y)]`, where `F` and `G` are normalized.
The resultant of `\phi` is the resultant of `F` and `G`.
Expand Down Expand Up @@ -908,8 +910,87 @@ def min_res_locus(self, ret_conjugation=False):
ALGORITHM:
See [Rum2013]_.
Algorithm A of [Rum2013]_ (p.28).
"""

# as step 2 of Rumely's algorithm may be run twice, we define it as a seperate function
def step_2(factorization, base):
min_list = []
for i in range(len(factorization)):
irreducible = factorization[i][0]
extension = base.extension(irreducible, 's%s'%i)
new_primes = extension.primes_above(base_prime_ideal)
for new_prime in new_primes:
#print('current prime = ', new_prime)
ramification_index = new_prime.absolute_ramification_index()
valuation = lambda x: x.valuation(new_prime)/ramification_index
s = extension.gens()[0]
transformation_matrix = Matrix([[1, s],[0, 1]])
new_system = system.change_ring(extension)
new_system = new_system.conjugate(transformation_matrix)
print('before normalizing coordinates = ', new_system)
new_system.normalize_coordinates() #very long call
#print('res = ', new_system.resultant())
res = valuation(new_system.resultant())
#print('ordres = ', res)
d = new_system.degree()
#print('d = ', d)
#print('system = ', new_system)
C_list = []
D_list = []
for i in range(d + 1):
C_i = new_system[0][i, d - i]
D_i = new_system[1][i, d - i]
#print('C_%s = ' %i, C_i)
#print('val = ', valuation(C_i))
if C_i != 0:
C_list.append(QQ(res - 2*d*valuation(C_i)))
else:
C_list.append(None)
if D_i != 0:
D_list.append(QQ(res - 2*d*valuation(D_i)))
else:
D_list.append(None)
#print('C_list =', C_list)
#print('D_list =', D_list)
D = QQ['t']
t = D.gens()[0]
C_lines = [(C_list[i] + QQ(d**2 + d - 2*d*i)*t) for i in range(len(C_list)) if C_list[i] != None]
D_lines = [(D_list[i] + QQ(d**2 + d - 2*d*(i + 1))*t) for i in range(len(D_list)) if D_list[i] != None]
#print('C_lines = ', C_lines)
#print('D_lines = ', D_lines)
all_lines = C_lines + D_lines #crude minimization
minimum = None
print('all lines = ', C_lines, D_lines)
all_intersections = []
for line_1 in all_lines:
for line_2 in all_lines:
if line_1 != line_2 and line_2[1] != line_1[1]:
intersection = QQ((line_1[0]-line_2[0])/(line_2[1]-line_1[1]))
if intersection not in all_intersections:
all_intersections.append(intersection)
#print(all_intersections)
for intersection in all_intersections:
C_max = max([(i(intersection)) for i in C_lines])
D_max = max([(i(intersection)) for i in D_lines])
print('intersection = ', intersection)
#print('C_max = ', C_max)
#print('D_max = ', D_max)
print('value of X(i) = ', max(C_max, D_max))
X_i = (max(C_max,D_max))
if X_i < minimum or minimum == None:
minimum = X_i
intersection_final = QQ(intersection)
print('minimum: ', minimum)
interval_check = False
for line in all_lines: #check if minimum is achieved on an interval
if line.is_constant():
if line[0] == minimum:
interval_check = True
break
min_list.append((minimum, intersection_final, interval_check, extension, new_prime))
return min_list

system = self._system
QQ_prime = self.domain().prime()
base_prime_ideal = self.domain().ideal()
Expand All @@ -924,88 +1005,16 @@ def min_res_locus(self, ret_conjugation=False):
num = phi(num)
#print('num:', num)
#print('dem:', dem)
fixed_point_equation = (num-w*dem)
fixed_point_equation = num - w*dem
factorization = list(fixed_point_equation.factor())
if(system([1,0]) == system.domain()([1,0])):
Q = dem
else:
value = system([1,0]).dehomogenize(1)[0]
Q = num - value*dem
factorization += list(Q.factor())
min_list = []
#print('all factors:', factorization)
for i in range(len(factorization)):
irreducible = factorization[i][0]
extension = base.extension(irreducible, 's%s'%i)
new_primes = extension.primes_above(base_prime_ideal)
for new_prime in new_primes:
#print('current prime = ', new_prime)
ramification_index = new_prime.absolute_ramification_index()
valuation = lambda x: x.valuation(new_prime)/ramification_index
s = extension.gens()[0]
transformation_matrix = Matrix([[1, s],[0, 1]])
new_system = system.change_ring(extension)
new_system = new_system.conjugate(transformation_matrix)
new_system.normalize_coordinates() #very long call
#print('res = ', new_system.resultant())
res = valuation(new_system.resultant())
#print('ordres = ', res)
d = new_system.degree()
#print('d = ', d)
#print('system = ', new_system)
C_list = []
D_list = []
for i in range(d+1):
C_i = new_system[0][i,d-i]
D_i = new_system[1][i,d-i]
#print('C_%s = ' %i, C_i)
#print('val = ', valuation(C_i))
if C_i != 0:
C_list.append(QQ(res-2*d*valuation(C_i)))
else:
C_list.append(None)
if D_i != 0:
D_list.append(QQ(res-2*d*valuation(D_i)))
else:
D_list.append(None)
#print('C_list =', C_list)
#print('D_list =', D_list)
D = QQ['t']
t = D.gens()[0]
C_lines = [(C_list[i] + QQ(d**2 + d - 2*d*i)*t) for i in range(len(C_list)) if C_list[i] != None]
D_lines = [(D_list[i] + QQ(d**2 + d - 2*d*(i + 1))*t) for i in range(len(D_list)) if D_list[i] != None]
#print('C_lines = ', C_lines)
#print('D_lines = ', D_lines)
all_lines = C_lines + D_lines #crude minimization
minimum = None
#print('all lines = ', C_lines, D_lines)
all_intersections = []
for line_1 in all_lines:
for line_2 in all_lines:
if line_1 != line_2 and line_2[1] != line_1[1]:
intersection = QQ((line_1[0]-line_2[0])/(line_2[1]-line_1[1]))
if intersection not in all_intersections:
all_intersections.append(intersection)
#print(all_intersections)
for intersection in all_intersections:
C_max = max([(i(intersection)) for i in C_lines])
D_max = max([(i(intersection)) for i in D_lines])
#print('intersection = ', intersection)
#print('C_max = ', C_max)
#print('D_max = ', D_max)
#print('value of X(i) = ', max(C_max, D_max))
X_i = (max(C_max,D_max))
if X_i < minimum or minimum == None:
minimum = (X_i)
intersection_final = QQ(intersection)
#print('minimum: ', minimum)
interval_check = False
for line in all_lines: #check if minimum is achieved on an interval
if line.is_constant():
if line[0] == minimum:
interval_check = True
break
min_list.append((minimum, intersection_final, interval_check, extension, new_prime))
min_list = step_2(factorization, base)
minimizing_list = [i[0] for i in min_list]
minimum = min(minimizing_list)
min_index = minimizing_list.index(minimum)
Expand All @@ -1025,32 +1034,26 @@ def min_res_locus(self, ret_conjugation=False):
if root == 1:
#print('in no denominator case')
#print('extension:', extension)
conj_matrix = Matrix([[QQ_prime**(intersection_final), extension.gens()[0]], [0,1]])
conj_matrix = Matrix([[QQ_prime**(intersection_final), extension.gens()[0]], [0, 1]])
new_system = system.change_ring(extension)
else:
C = extension['c']
c = C.gens()[0]
defining_polynomial = c**(root) - QQ_prime**(intersection_final.numerator())
factorization = defining_polynomial.factor()
degree = None
for factor in factorization:
if degree == None or factor[0].degree() < degree: #slow symbolic calcuation
final_factor = factor[0]
degree = factor[0].degree()
if degree != 1:
new_extension = extension.extension(final_factor, 'w0')
A = new_extension.gens()[0]
B = new_extension.gens()[1]
else:
A = final_factor[0]
B = extension.gens()[0]
conj_matrix = Matrix([[A, B],[0,1]])
defining_polynomial = c**root - QQ_prime**intersection_final.numerator()
#print('defining polynomial = ', defining_polynomial)
new_extension, ext_to_new_ext = defining_polynomial.splitting_field('w0', map=True)
A = defining_polynomial.roots(new_extension)[0][0]
B = ext_to_new_ext(extension.gens()[0])
conj_matrix = Matrix([[A, B], [0, 1]])
new_system = system.change_ring(new_extension)
new_system = new_system.conjugate(conj_matrix)
return (min_res_locus, min_ord_res, conj_matrix)
return (min_res_locus, min_ord_res, conj_matrix, new_extension, extension, ext_to_new_ext)
else:
print('not implemented')
return [min_list[min_index][0], min_list[min_index][1]]
list_of_minimums = [tup for tup in min_list if tup[0] == min_ord_res]
max_radius = max([QQ_prime**(-1*tup[1]) for tup in list_of_minimums])
maximum_radii = [tup for tup in list_of_minimums if QQ_prime**(-1*tup[1]) == max_radius]
return maximum_radii


class DynamicalSystem_Berkovich_affine(DynamicalSystem_Berkovich):
r"""
Expand Down

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