diffn for arbitrary template should work
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@ -1,6 +1,7 @@
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class as_cover:
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def __init__(self, C, cover_template, list_of_fcts, branch_points = [], prec = 10):
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self.quotient = C
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self.cover_template = cover_template
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self.functions = list_of_fcts
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print('a')
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self.height = len(list_of_fcts)
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@ -77,17 +78,38 @@ class as_cover:
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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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Rzf, zgen, fgen = cover_template.fct_field
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subs_fs = {zgen[i] : z[i]}| {fgen[i] : RxyzQ(list_of_fcts[i].function) for i in range(n)}
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self.rhs = [RxyzQ(cover_template.fcts[i].subs(subs_fs)) for i in range(n)]
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#####
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##### We compute now the differentials dz[i]
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y_super = superelliptic_function(C, y)
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dy_super = y_super.diffn().form
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dz = []
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for i in range(n):
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aux_fct = self.rhs[i]
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result = 0
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for j in range(i):
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result += aux_fct.derivative(z[j])*dz[j]
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result += aux_fct.derivative(x)
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result += aux_fct.derivative(y)*dy_super
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dz += [-result]
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self.dz = dz
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def __repr__(self):
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n = self.height
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p = self.characteristic
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if n==1:
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return "(Z/p)-cover of " + str(self.quotient)+" with the equation:\n z^" + str(p) + " - z = " + str(self.functions[0])
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result = "(Z/p)^"+str(self.height)+ "-cover of " + str(self.quotient)+" with the equations:\n"
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result = str(self.group)
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result += "-cover of " + str(self.quotient)+" with the equations: \n"
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for i in range(n):
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result += 'z' + str(i) + "^" + str(p) + " - z" + str(i) + " = " + str(self.functions[i]) + "\n"
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result += 'z'+str(i)+'^p - z'+str(i) + ' = '
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aux = str(self.cover_template.fcts[i])
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for t in range(n):
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aux = aux.replace("f"+str(t), "(" + str(self.functions[t]) + ")")
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result += aux + '\n'
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return result
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def genus(self):
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@ -289,7 +311,7 @@ class as_cover:
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def ith_ramification_gp(self, i, place = 0):
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'''Find ith ramification group at place at infty of nb place.'''
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G = self.group
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G = self.group.elts
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t = self.uniformizer(place)
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Gi = [G[0]]
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for g in G:
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@ -302,7 +324,7 @@ class as_cover:
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def ramification_jumps(self, place = 0):
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'''Return list of lower ramification jumps at at place at infty of nb place.'''
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G = self.group
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G = self.group.elts
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ramification_jps = []
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i = 0
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while len(G) > 1:
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@ -3,13 +3,7 @@ class as_function:
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self.curve = C
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F = C.base_ring
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n = C.height
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variable_names = 'x, y'
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for i in range(n):
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variable_names += ', z' + str(i)
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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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RxyzQ, Rxyz, x, y, z = C.fct_field
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self.function = RxyzQ(g)
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#self.function = as_reduction(AS, RxyzQ(g))
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@ -81,13 +75,7 @@ class as_function:
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y_series = C.y_series[place]
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z_series = C.z_series[place]
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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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RxyzQ, Rxyz, x, y, z = C.fct_field
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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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@ -103,13 +91,7 @@ class as_function:
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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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RxyzQ, Rxyz, x, y, z = C.fct_field
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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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@ -117,37 +99,39 @@ class as_function:
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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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def group_action(self, elt):
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C = self.curve
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n = C.height
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F = C.base_ring
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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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sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}
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g = self.function
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return as_function(C, g.substitute(sub_list))
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RxyzQ, Rxyz, x, y, z = C.fct_field
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Rzf, zgen, fgen = C.cover_template.fct_field
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if isinstance(elt, group_elt):
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elt = elt.as_tuple
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AS = self.curve
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n = AS.height
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G = AS.group
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if elt in G.gens:
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ind = G.gens.index(elt)
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gp_action_list = C.cover_template.gp_action[ind]
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sub_list = {x : x, y : y} | {z[j] : Rxyz(gp_action_list[j].subs({zgen[i] : z[i] for i in range(n)})) for j in range(n)}
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g = self.function
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return as_function(C, g.substitute(sub_list))
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result = self
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for i in range(n):
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for j in range(elt[i]):
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result = result.group_action(G.gens[i])
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return result
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def reduce(self):
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aux = as_reduction(self.curve, self.function)
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return as_function(self.curve, aux)
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def trace(self):
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def trace(self, super=True):
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C = self.curve
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C_super = C.quotient
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n = C.height
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F = C.base_ring
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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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RxyzQ, Rxyz, x, y, z = C.fct_field
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result = as_function(C, 0)
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G = C.group
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for a in G:
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@ -156,7 +140,10 @@ class as_function:
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Rxy.<x, y> = PolynomialRing(F, 2)
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Qxy = FractionField(Rxy)
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result = as_reduction(C, result)
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return superelliptic_function(C_super, Qxy(result))
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if super:
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return superelliptic_function(C_super, Qxy(result))
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return as_function(AS, Qxy(result))
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def coordinates(self, prec = 100, basis = 0):
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"Return coordinates in H^1(X, OX)."
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@ -180,14 +167,10 @@ class as_function:
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C_super = C.quotient
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n = C.height
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RxyzQ, Rxyz, x, y, z = C.fct_field
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fcts = C.functions
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f = self.function
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y_super = superelliptic_function(C_super, y)
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dy_super = y_super.diffn().form
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dz = []
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for i in range(n):
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dfct = fcts[i].diffn().form
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dz += [-dfct]
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dz = C.dz
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result = f.derivative(x)
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result += f.derivative(y)*dy_super
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for i in range(n):
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