as symmetric power basis
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@ -135,11 +135,11 @@ class as_cover:
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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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forms = as_holomorphic_combinations(S)
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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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forms = as_holomorphic_combinations(forms)
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if len(forms) < self.genus():
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print("I haven't found all forms, only ", len(forms), " of ", self.genus())
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@ -376,7 +376,7 @@ class as_cover:
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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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S += [(eta, eta_exp)]
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forms = holomorphic_combinations(S)
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forms = as_holomorphic_combinations(S)
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if len(forms) <= self.genus():
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raise ValueError("Increase threshold!")
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for omega in forms:
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@ -398,7 +398,7 @@ class as_cover:
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result += [as_cech(self, omega, f)]
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return result
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def holomorphic_combinations(S):
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def as_holomorphic_combinations(S):
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"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
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C_AS = S[0][0].curve
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p = C_AS.characteristic
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@ -8,7 +8,7 @@ class as_polyform:
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return as_polyform(self.form + other.form, self.mult)
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def __repr__(self):
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return str(self.form) + ' dx⊗' + str(self.mult)
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return '(' + str(self.form) + ') dx⊗' + str(self.mult)
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def expansion_at_infty(self):
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return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
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@ -30,39 +30,76 @@ class as_polyform:
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def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
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v = AS.dx.valuation()
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result = AS.at_most_poles(mult*v, threshold=threshold)
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return [as_polyform(omega, mult) for omega in result]
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result = [as_polyform(omega, mult) for omega in result]
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if mult == 1 and len(result) < AS.genus():
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raise ValueError('Increase threshold, not all forms found.')
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if mult > 1 and len(result) < (2*mult - 1)*(AS.genus() - 1):
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raise ValueError('Increase threshold, not all forms found.')
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return result
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as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis
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def as_canonical_ideal(AS, threshold=8):
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def as_symmetric_power_basis(AS, n, threshold = 8):
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g = AS.genus()
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B0 = AS.holomorphic_differentials_basis(threshold=threshold)
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from itertools import product
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indices = [list(range(g)) for i in range(n)]
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indices_nonrepeating = []
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for i in product(*indices):
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if non_decreasing(i):
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indices_nonrepeating += [i]
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result = []
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for i in indices_nonrepeating:
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tensor_form = [as_function(AS, B0[i[j]].form) for j in range(n)]
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tensor_form2 = [B0[i[j]] for j in range(n)]
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print(hash(tuple(tensor_form)))
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print([tensor_form], tuple(tensor_form))
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aux_dict = {}
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aux_dict[tuple(list(tensor_form))] = 1
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print(aux_dict)
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result += [as_symmetric_product_forms([tensor_form], aux_dict)]
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print(binomial(g + n - 1, n), len(result))
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return result
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def as_canonical_ideal(AS, n, threshold=8):
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B0 = AS.holomorphic_differentials_basis(threshold=threshold)
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F = AS.base_ring
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g = AS.genus()
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B1 = [(B0[i], B0[j]) for i in range(g) for j in range(g) if i <= j]
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B2 = AS.holo_polydifferentials_basis(2, threshold = threshold)
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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from itertools import product
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#B1 = as_symmetric_power_basis(AS, n, threshold = 8)
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B1 = [[B0[i[j]] for j in range(n)] for i in product(*indices)]
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B2 = AS.holo_polydifferentials_basis(n, threshold = threshold)
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g = AS.genus()
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r = len(B2)
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M = matrix(F, g^2, r)
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M = matrix(F, len(B1), r)
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for i in range(0, len(B1)):
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(a, b) = B1[i]
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c = as_function(AS, a.form*b.form)
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atuple = B1[i]
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c = Rxyz(1)
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for a in atuple:
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c = c*a.form
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c = as_reduction(AS, c)
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c = as_function(AS, c)
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c = as_polyform(c, 2)
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c = as_polyform(c, n)
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#return M, c.coordinates(basis=B2)
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M[i, :] = vector(c.coordinates(basis=B2))
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K = M.kernel().basis()
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result = []
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for v in K:
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coeffs = {b : 0 for b in B1}
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coeffs = {tuple(b) : 0 for b in B1}
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for i in range(r):
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if v[i] != 0:
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coeffs[B1[i]] += v[i]
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result += [as_tensor_product_forms(B1, coeffs)]
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coeffs[tuple(B1[i])] += v[i]
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result += [as_symmetric_product_forms(B1, coeffs)]
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return result
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as_cover.canonical_ideal = as_canonical_ideal
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def as_canonical_ideal_polynomials(AS, n, threshold=8):
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return [a.polynomial() for a in AS.canonical_ideal(n, threshold=threshold)]
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as_cover.canonical_ideal_polynomials = as_canonical_ideal_polynomials
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class as_tensor_product_forms:
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def __init__(self, pairs_of_forms, coeffs):
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self.pairs = pairs_of_forms
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@ -106,4 +143,83 @@ class as_tensor_product_forms:
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elt1 = [p^n - a for a in elt]
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pairs_of_forms2 = [(a.group_action(elt), b.group_action(elt1)) for (a, b) in pairs_of_forms]
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coeffs2 = {(a.group_action(elt), b.group_action(elt1)) : self.coeffs[(a, b)] for (a, b) in pairs_of_forms}
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return as_tensor_product_forms(pairs_of_forms2, self.coeffs)
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return as_tensor_product_forms(pairs_of_forms2, self.coeffs)
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class as_symmetric_product_forms:
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def __init__(self, tuples_of_forms, coeffs):
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self.n = len(tuples_of_forms[0])
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self.tuples = tuples_of_forms
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self.coeffs = coeffs #dictionary
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self.curve = tuples_of_forms[0][0].curve
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def coordinates(self, basis = 0):
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AS = self.curve
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g = AS.genus()
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F = AS.base_ring
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if basis == 0:
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basis = AS.holomorphic_differentials_basis()
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from itertools import product
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indices = [list(range(g)) for i in range(self.n)]
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result = {i : 0 for i in product(*indices)}
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for atuple in self.tuples:
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coors = [omega.coordinates(basis = basis) for omega in atuple]
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for i in product(*indices):
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aux_product = 1
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for j in range(n):
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aux_product *= coors[j][i[j]]
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result[i] += self.coeffs[tuple(atuple)]*aux_product
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return result
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def __repr__(self):
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result = ''
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for atuple in self.tuples:
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if self.coeffs[tuple(atuple)] !=0:
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result += str(self.coeffs[tuple(atuple)]) + ' * '
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for j in range(self.n):
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result += "(" + str(atuple[j]) + ")"
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if j != self.n-1:
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result + '⊗'
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result += ' + '
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return result
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def multiply(self):
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n = self.n
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AS = self.curve
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RxyzQ, Rxyz, x, y, z = AS.fct_field
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result = as_polyform(0*AS.x, n)
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for atuple in self.tuples:
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aux_product = Rxyz(1)
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for fct in atuple:
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aux_product = aux_product * fct.form
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aux_product = as_function(AS, aux_product)
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result += as_polyform(self.coeffs[tuple(atuple)]*aux_product, n)
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return result
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def polynomial(self):
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AS = self.curve
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F = AS.base_ring
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g = AS.genus()
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M = self.coordinates()
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n = self.n
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Rg = PolynomialRing(F, 'X', g)
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X = Rg.gens()
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from itertools import product
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indices = [list(range(g)) for i in range(self.n)]
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result = Rg(0)
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for i in product(*indices):
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aux_product = Rg(1)
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for j in range(n):
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aux_product *= X[i[j]]
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result += M[i] * aux_product
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return result
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def group_action(self, elt):
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p = self.base_ring.characteristic()
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n = self.height
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elt1 = [p^n - a for a in elt]
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pairs_of_forms2 = [(a.group_action(elt), b.group_action(elt1)) for (a, b) in pairs_of_forms]
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coeffs2 = {(a.group_action(elt), b.group_action(elt1)) : self.coeffs[(a, b)] for (a, b) in pairs_of_forms}
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return as_tensor_product_forms(pairs_of_forms2, self.coeffs)
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def non_decreasing(L):
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return all(x<=y for x, y in zip(L, L[1:]))
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@ -137,11 +137,11 @@ class quaternion_cover:
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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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forms = quaternion_holomorphic_combinations(S)
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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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forms = quaternion_holomorphic_combinations(forms)
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if len(forms) < self.genus():
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print("I haven't found all forms, only ", len(forms), " of ", self.genus())
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@ -378,7 +378,7 @@ class quaternion_cover:
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eta = quaternion_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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S += [(eta, eta_exp)]
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forms = holomorphic_combinations(S)
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forms = quaternion_holomorphic_combinations(S)
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if len(forms) <= self.genus():
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raise ValueError("Increase threshold!")
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for omega in forms:
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@ -400,7 +400,7 @@ class quaternion_cover:
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result += [quaternion_cech(self, omega, f)]
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return result
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def holomorphic_combinations(S):
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def quaternion_holomorphic_combinations(S):
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"""Given a list S of pairs (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt."""
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C_AS = S[0][0].curve
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p = C_AS.characteristic
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