nieudane proby coordinates dla cech_drw
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sage/.run.term-0.term
41001
sage/.run.term-0.term
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@ -1,5 +1,5 @@
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class as_cover:
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def __init__(self, C, list_of_fcts, prec = 10):
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def __init__(self, C, list_of_fcts, branch_points = [], prec = 10):
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self.quotient = C
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self.functions = list_of_fcts
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self.height = len(list_of_fcts)
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@ -22,24 +22,25 @@ class as_cover:
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r = f.degree()
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delta = GCD(m, r)
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self.nb_of_pts_at_infty = delta
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self.branch_points = list(range(delta)) + branch_points
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Rxy.<x, y> = PolynomialRing(F, 2)
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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all_x_series = []
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all_y_series = []
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all_z_series = []
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all_dx_series = []
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all_jumps = []
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all_x_series = {}
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all_y_series = {}
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all_z_series = {}
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all_dx_series = {}
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all_jumps = {}
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for i in range(delta):
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x_series = superelliptic_function(C, x).expansion_at_infty(place = i, prec=prec)
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y_series = superelliptic_function(C, y).expansion_at_infty(place = i, prec=prec)
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for pt in self.branch_points:
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x_series = superelliptic_function(C, x).expansion(pt=pt, prec=prec)
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y_series = superelliptic_function(C, y).expansion(pt=pt, prec=prec)
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z_series = []
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jumps = []
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n = len(list_of_fcts)
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list_of_power_series = [g.expansion_at_infty(place = i, prec=prec) for g in list_of_fcts]
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for i in range(n):
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power_series = list_of_power_series[i]
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list_of_power_series = [g.expansion(pt=pt, prec=prec) for g in list_of_fcts]
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for j in range(n):
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power_series = list_of_power_series[j]
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jump, correction, t_old, z = artin_schreier_transform(power_series, prec = prec)
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x_series = x_series(t = t_old)
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y_series = y_series(t = t_old)
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@ -48,11 +49,11 @@ class as_cover:
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jumps += [jump]
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list_of_power_series = [g(t = t_old) for g in list_of_power_series]
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all_jumps += [jumps]
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all_x_series += [x_series]
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all_y_series += [y_series]
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all_z_series += [z_series]
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all_dx_series += [x_series.derivative()]
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all_jumps[pt] = jumps
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all_x_series[pt] = x_series
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all_y_series[pt] = y_series
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all_z_series[pt] = z_series
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all_dx_series[pt] = x_series.derivative()
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self.jumps = all_jumps
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self.x_series = all_x_series
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self.y_series = all_y_series
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@ -70,8 +71,9 @@ class as_cover:
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self.fct_field = (RxyzQ, Rxyz, x, y, z)
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self.x = as_function(self, x)
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self.y = as_function(self, y)
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self.z = [as_function(self, z[i]) for i in range(n)]
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self.z = [as_function(self, z[j]) for j in range(n)]
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self.dx = as_form(self, 1)
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self.one = as_function(self, 1)
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def __repr__(self):
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@ -89,9 +91,9 @@ class as_cover:
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jumps = self.jumps
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gY = self.quotient.genus()
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n = self.height
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delta = self.nb_of_pts_at_infty
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branch_pts = self.branch_points
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p = self.characteristic
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return p^n*gY + (p^n - 1)*(delta - 1) + sum(p^(n-j-1)*(jumps[i][j]-1)*(p-1)/2 for j in range(n) for i in range(delta))
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return p^n*gY + (p^n - 1)*(len(branch_pts) - 1) + sum(p^(n-j-1)*(jumps[pt][j]-1)*(p-1)/2 for j in range(n) for pt in branch_pts)
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def exponent_of_different(self, place = 0):
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jumps = self.jumps
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@ -130,17 +132,17 @@ class as_cover:
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for j in range(0, m):
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for k in product(*pr):
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eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)
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eta_exp = eta.expansion_at_infty()
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eta_exp = eta.expansion(pt=self.branch_points[0])
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S += [(eta, eta_exp)]
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forms = holomorphic_combinations(S)
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for i in range(1, delta):
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forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
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for pt in self.branch_points[1:]:
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forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
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forms = holomorphic_combinations(forms)
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if len(forms) < self.genus():
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print("I haven't found all forms.")
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print("I haven't found all forms, only ", len(forms), " of ", self.genus())
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return holomorphic_differentials_basis(self, threshold = threshold + 1)
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if len(forms) > self.genus():
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print("Increase precision.")
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@ -236,8 +238,8 @@ class as_cover:
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forms = holomorphic_combinations_forms(S, pole_order)
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for i in range(1, delta):
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forms = [(omega, omega.expansion_at_infty(place = i)) for omega in forms]
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for pt in self.branch_points[1:]:
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forms = [(omega, omega.expansion(pt=pt)) for omega in forms]
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forms = holomorphic_combinations_forms(forms, pole_order)
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return forms
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@ -37,6 +37,28 @@ class as_form:
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sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
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return g.substitute(sub_list)*dx_series
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def expansion(self, pt = 0):
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'''Same code as expansion_at_infty.'''
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C = self.curve
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F = C.base_ring
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x_series = C.x_series[pt]
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y_series = C.y_series[pt]
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z_series = C.z_series[pt]
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dx_series = C.dx_series[pt]
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n = C.height
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variable_names = 'x, y'
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for j in range(n):
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variable_names += ', z' + str(j)
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Rxyz = PolynomialRing(F, n+2, variable_names)
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x, y = Rxyz.gens()[:2]
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z = Rxyz.gens()[2:]
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RxyzQ = FractionField(Rxyz)
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prec = C.prec
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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g = self.form
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sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
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return g.substitute(sub_list)*dx_series
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def __add__(self, other):
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C = self.curve
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g1 = self.form
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@ -152,6 +174,13 @@ def artin_schreier_transform(power_series, prec = 10):
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power_series = RtQ(power_series)
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if power_series.valuation() == +Infinity:
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raise ValueError("Precision is too low.")
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if power_series.valuation() >= 0:
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# THIS IS WRONG - THERE ARE SEVERAL PLACES OVER THIS PLACE, AND IT DEPENDS
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aux = t^p - t
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z = new_reverse(aux, prec = prec)
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z = z(t = power_series)
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return(0, 0, t, z)
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while(power_series.valuation() % p == 0 and power_series.valuation() < 0):
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M = -power_series.valuation()/p
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coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS
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@ -77,6 +77,28 @@ class as_function:
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g = RxyzQ(g)
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sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
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return g.substitute(sub_list)
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def expansion(self, pt = 0):
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C = self.curve
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delta = C.nb_of_pts_at_infty
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F = C.base_ring
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x_series = C.x_series[pt]
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y_series = C.y_series[pt]
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z_series = C.z_series[pt]
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n = C.height
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variable_names = 'x, y'
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for j in range(n):
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variable_names += ', z' + str(j)
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Rxyz = PolynomialRing(F, n+2, variable_names)
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x, y = Rxyz.gens()[:2]
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z = Rxyz.gens()[2:]
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RxyzQ = FractionField(Rxyz)
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prec = C.prec
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Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
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g = self.function
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g = RxyzQ(g)
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sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}
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return g.substitute(sub_list)
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def group_action(self, ZN_tuple):
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C = self.curve
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@ -2,13 +2,15 @@ p = 3
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m = 2
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F = GF(p)
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Rx.<x> = PolynomialRing(F)
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f = x^3 - x + 1
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f = x^3 - x
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C = superelliptic(f, m)
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C1 = patch(C)
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#C1 = patch(C)
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#print(C1.crystalline_cohomology_basis())
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g1 = C1.polynomial
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g_AS = g1(x^p - x)
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C2 = superelliptic(g_AS, 2)
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print(convert_super_into_AS(C2))
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converted = (C2.x)^4 - (C2.x)^2
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print(convert_super_fct_into_AS(converted))
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#g1 = C1.polynomial
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#g_AS = g1(x^p - x)
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#C2 = superelliptic(g_AS, 2)
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#print(convert_super_into_AS(C2))
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#converted = (C2.x)^4 - (C2.x)^2
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#print(convert_super_fct_into_AS(converted))
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b = C.crystalline_cohomology_basis()
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print(autom(b[0]).coordinates(basis = b))
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@ -47,10 +47,11 @@ class superelliptic_witt:
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return other_integer*self
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def __mul__(self, other):
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C = self.curve
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p = C.characteristic
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if isinstance(other, superelliptic_witt):
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t1 = self.t
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f1 = self.f
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p = self.curve.characteristic
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t2 = other.t
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f2 = other.f
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return superelliptic_witt(t1*t2, t1^p*f2 + t2^p*f1)
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@ -58,7 +59,6 @@ class superelliptic_witt:
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h1 = other.h1
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h2 = other.h2
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omega = other.omega
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p = other.curve.characteristic
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t = self.t
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f = self.f
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aux_form = t^p*omega - h2*t^(p-1)*t.diffn() + f*h1^p*(C.x)^(p-1)*C.dx
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@ -130,7 +130,7 @@ superelliptic_function.teichmuller = teichmuller
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#d[M] = a*[b x^(a-1)]
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def auxilliary_derivative(P):
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'''Return "derivative" of P, where P depends only on x. '''
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'''Return "derivative" of P, where P depends only on x. In other words d[P(x)].'''
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P0 = P.t.function
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P1 = P.f.function
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C = P.curve
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@ -156,6 +156,11 @@ class superelliptic_drw_form:
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self.omega = omega
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self.h2 = h2
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def r(self):
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C = self.curve
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h1 = self.h1
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return superelliptic_form(C, h1.function)
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def __eq__(self, other):
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eq1 = (self.h1 == self.h1)
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try:
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@ -317,16 +322,29 @@ class superelliptic_drw_cech:
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return superelliptic_cech(C, omega0.h1*C.dx, f.t)
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def coordinates(self, basis = 0):
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C = self.curve
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g = C.genus()
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coord_mod_p = self.r().coordinates()
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print(coord_mod_p)
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coord_lifted = [lift(a) for a in coord_mod_p]
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if basis == 0:
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basis = self.curve().crystalline_cohomology_basis()
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basis = C.crystalline_cohomology_basis()
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aux = self
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for i, a in enumerate(basis):
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aux -= coord_lifted[i]*a
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aux = aux.reduce()
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return aux
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print('aux before reduce', aux)
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#aux = aux.reduce() # Now aux = p*cech class.
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h02 = aux.omega0.h2
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h82 = aux.omega8.h2
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aux -= superelliptic_drw_cech(h02.verschiebung().diffn(), (h02 - h82).verschiebung())
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# Now aux should be of the form (V(smth), V(smth), V(smth))
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print('aux V(smth)', aux)
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aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
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print('aux.omega0.omega.cartier()', aux.omega0.omega.cartier())
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coord_aux_divided_by_p = aux_divided_by_p.coordinates()
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coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
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coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
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return coordinates
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def de_rham_witt_lift(cech_class, prec = 50):
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@ -335,19 +353,24 @@ def de_rham_witt_lift(cech_class, prec = 50):
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omega0 = cech_class.omega0
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omega8 = cech_class.omega8
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fct = cech_class.f
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omega0_regular = regular_form(omega0)
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omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
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print(omega0_regular)
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omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
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omega8_regular = regular_form(second_patch(omega8))
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#Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
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omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
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omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
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u = (C.x)^(-1)
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v = (C.y)/(C.x)^(g+1)
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omega8_lift = omega0_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(v.teichmuller().diffn())
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aux = omega0_lift - omega8_lift - fct.teichmuller().diffn()
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decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec)
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omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
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print('omega8_lift.frobenius().expansion_at_infty()', omega8_lift.frobenius().expansion_at_infty())
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aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
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if aux.h1.function != 0:
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raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
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decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec) #decompose dV(smth) in aux as smth regular on U0 - smth regular on U8.
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aux_h2 = decom_aux_h2[0]
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aux_f = decom_aux_h2[2]
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aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
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result = superelliptic_drw_cech(omega0_lift + aux_h2.verschiebung().diffn() + aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
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result = superelliptic_drw_cech(omega0_lift - aux_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
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return result.reduce()
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def crystalline_cohomology_basis(self, prec = 50):
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@ -25,6 +25,7 @@ load('drafty/regular_on_U0.sage')
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load('drafty/lift_to_de_rham.sage')
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#load('drafty/superelliptic_cohomology_class.sage')
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load('drafty/superelliptic_drw.sage')
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load('drafty/draft.sage')
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#load('drafty/draft_klein_covers.sage')
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load('drafty/2gpcovers.sage')
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#load('drafty/2gpcovers.sage')
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load('drafty/pole_numbers.sage')
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@ -1,5 +1,6 @@
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def decomposition_g0_g8(fct, prec = 50):
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'''Writes fct as a difference g0 - g8, with g0 regular on the affine patch and g8 at the points in infinity.'''
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'''Writes fct as a difference g0 - g8 + f, with g0 regular on the affine patch and g8 at the points in infinity
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and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
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C = fct.curve
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g = C.genus()
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coord = fct.coordinates()
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@ -7,8 +8,6 @@ def decomposition_g0_g8(fct, prec = 50):
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for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
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nontrivial_part += coord[i]*a
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fct -= nontrivial_part
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if fct.coordinates(prec=prec) != g*[0]:
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raise ValueError("The given function cannot be written as g0 - g8.")
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Fxy, Rxy, x, y = C.fct_field
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fct = Fxy(fct.function)
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@ -20,7 +19,7 @@ def decomposition_g0_g8(fct, prec = 50):
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for monomial in num.monomials():
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aux = superelliptic_function(C, monomial)
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if aux.expansion_at_infty().valuation() >= aux_den.expansion_at_infty().valuation():
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g8 += num.monomial_coefficient(monomial)*aux/aux_den
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g8 -= num.monomial_coefficient(monomial)*aux/aux_den
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else:
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g0 += num.monomial_coefficient(monomial)*aux/aux_den
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return (g0, g8, nontrivial_part)
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@ -100,6 +100,8 @@ class superelliptic_form:
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FxRy.<y> = PolynomialRing(Fx)
|
||||
g = reduction(C, y^m*g)
|
||||
g = FxRy(g)
|
||||
if j == 0:
|
||||
return g.monomial_coefficient(y^(0))/C.polynomial
|
||||
return g.monomial_coefficient(y^(m-j))
|
||||
|
||||
def is_regular_on_U0(self):
|
||||
@ -107,7 +109,7 @@ class superelliptic_form:
|
||||
F = C.base_ring
|
||||
m = C.exponent
|
||||
Rx.<x> = PolynomialRing(F)
|
||||
for j in range(1, m):
|
||||
for j in range(0, m):
|
||||
if self.jth_component(j) not in Rx:
|
||||
return 0
|
||||
return 1
|
||||
@ -138,6 +140,15 @@ class superelliptic_form:
|
||||
dx_series = x_series.derivative()
|
||||
return g*dx_series
|
||||
|
||||
def expansion(self, pt, prec = 50):
|
||||
'''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''
|
||||
if pt in ZZ:
|
||||
return self.expansion_at_infty(place=pt, prec=prec)
|
||||
C = self.curve
|
||||
dx_series = C.x.expansion(pt = pt, prec=prec).derivative()
|
||||
aux_fct = superelliptic_function(C, self.form)
|
||||
return aux_fct.expansion(pt=pt, prec=prec)*dx_series
|
||||
|
||||
def residue(self, place = 0, prec=30):
|
||||
return self.expansion_at_infty(place = place, prec=prec)[-1]
|
||||
|
||||
|
@ -135,6 +135,30 @@ class superelliptic_function:
|
||||
xx = Rt(1/(t^M*ww^b))
|
||||
yy = 1/(t^R*ww^a)
|
||||
return Rt(fct(x = Rt(xx), y = Rt(yy)))
|
||||
|
||||
def expansion(self, pt, prec = 50):
|
||||
'''Expansion in the completed ring of the point pt. If pt is an integer, it means the corresponding place at infinity.'''
|
||||
if pt in ZZ:
|
||||
return self.expansion_at_infty(place=pt, prec=prec)
|
||||
x0, y0 = pt[0], pt[1]
|
||||
C = self.curve
|
||||
f = C.polynomial
|
||||
F = C.base_ring
|
||||
m = C.exponent
|
||||
Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
|
||||
Rxy.<x, y> = PolynomialRing(F, 2)
|
||||
Fxy = FractionField(Rxy)
|
||||
if y0 !=0 and f.derivative()(x0) != 0:
|
||||
y_series = f(x = t + x0).nth_root(m)
|
||||
return Rt(self.function(x = t + x0, y = y_series))
|
||||
if f.derivative()(x0) == 0: # then x - x0 is a uniformizer
|
||||
y_series = Rt(f(x = t+x0).nth_root(m))
|
||||
return Rt(self.function(x = t + x0, y = y_series))
|
||||
if y0 == 0: #then y is a uniformizer
|
||||
f1 = f(x = x+x0) - y0
|
||||
x_series = new_reverse(f1(x = t), prec = prec)
|
||||
x_series = x_series(t = t^m - y0) + x0
|
||||
return self.function(x = x_series, y = t)
|
||||
|
||||
def pth_root(self):
|
||||
'''Compute p-th root of given function. This uses the following fact: if h = H^p, then C(h*dx/x) = H*dx/x.'''
|
||||
|
Loading…
Reference in New Issue
Block a user