as reduction (stare) plus macierz dzialania na bazie dR prawie dziala

sage/as_covers/as_cech_class.sage

@@ -36,4 +36,40 @@ class as_cech:
         C = self.curve
         omega = self.omega0
         f = self.f
-        return as_form(C, constant*omega, constant*f)
+        return as_cech(C, constant*omega, constant*f)
+    
+    def coordinates(self, threshold=10, basis = 0):
+        '''Find coordinates of self in the de Rham cohomology basis. Threshold is an argument passed to AS.de_rham_basis().'''
+        AS = self.curve
+        if basis == 0:
+            basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
+        holo_diffs = basis[0]
+        coh_basis =  basis[1]
+        dR = basis[2]
+        F = AS.base_ring
+        f_products = []
+        for i, f in enumerate(coh_basis):
+            f_products += [[]]
+            for omega in holo_diffs:
+                f_products[i] += [sum((f*omega).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))]
+
+        product_of_fct_and_omegas = []
+        fct = self.f
+        for omega in holo_diffs:
+                product_of_fct_and_omegas += [sum((fct*omega).residue(place = _) for _ in range(AS.nb_of_pts_at_infty))]
+
+        V = (F^(AS.genus())).span_of_basis([vector(a) for a in f_products])
+        coh_coordinates = V.coordinates(product_of_fct_and_omegas) #coeficients of self in the basis elts coming from cohomology of OX
+        for i in range(AS.genus()):
+            self -= coh_coordinates[i]*dR[i+AS.genus()]
+
+        hol_form = self.omega0 + self.f.diffn() #now this should be a holomorphic form
+        hol_form = hol_form
+        print(hol_form)
+        return hol_form.coordinates() + coh_coordinates
+    
+    def group_action(self, g):
+        AS = self.curve
+        omega = self.omega0
+        f = self.f
+        return as_cech(self.curve, omega.group_action(g), f.group_action(g))

sage/as_covers/as_cover_class.sage

@@ -370,7 +370,7 @@ class as_cover:
     def de_rham_basis(self, threshold = 30):
         result = []
         for omega in self.holomorphic_differentials_basis():
-            result += [(omega, 0)]
+            result += [as_cech(self, omega, as_function(self, 0))]
         for f in self.cohomology_of_structure_sheaf_basis():
             omega = self.lift_to_de_rham(f, threshold = threshold)
             result += [as_cech(self, omega, f)]

sage/as_covers/as_form_class.sage

@@ -69,19 +69,18 @@ class as_form:
         g = self.form
         return as_form(C, g.substitute(sub_list))
 
-    def coordinates(self, holo):
-        """Find coordinates of the given form self in terms of the basis forms in a list holo."""
+    def coordinates(self, basis = 0):
+        """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
         C = self.curve
-        n = C.height
-        gC = C.genus()
-        F = C.base_ring
-        variable_names = 'x, y'
-        for j in range(n):
-            variable_names += ', z' + str(j)
-        Rxyz = PolynomialRing(F, n+2, variable_names)
-        x, y = Rxyz.gens()[:2]
-        z = Rxyz.gens()[2:]
-        return linear_representation_polynomials(Rxyz(self.form), holo)
+        if basis == 0:
+            basis = C.holomorphic_differentials_basis()
+        RxyzQ, Rxyz, x, y, z = C.fct_field
+        # We need to have only polynomials to use monomial_coefficients in linear_representation_polynomials,
+        # and sometimes basis elements have denominators. Thus we multiply by them.
+        denom = LCM([denominator(omega.form) for omega in basis])
+        basis = [denom*omega for omega in basis]
+        self_with_no_denominator = denom*self
+        return linear_representation_polynomials(Rxyz(self_with_no_denominator.form), [Rxyz(omega.form) for omega in basis])
 
     def trace(self):
         C = self.curve

sage/as_covers/as_reduction.sage

@@ -0,0 +1,32 @@
+def as_reduction(AS, fct):
+    '''Simplify rational function fct as a function in the function field of AS, so that z[i] appear in powers <p and only in numerator'''
+    n = AS.height
+    F = AS.base_ring
+    RxyzQ, Rxyz, x, y, z = AS.fct_field
+    ff = AS.functions
+    ff = [RxyzQ(F.function) for F in ff]
+    fct = RxyzQ(fct)
+    fct1 = numerator(fct)
+    fct2 = denominator(fct)
+    denom = as_function(AS, fct2)
+    denom_norm = prod(as_function(AS, fct2).group_action(g) for g in AS.group if list(g) != n*[0])
+    fct1 = Rxyz(fct1*denom_norm.function)
+    fct2 = Rxyz(fct2*denom_norm.function)
+    if fct2 != 1:
+        return as_reduction(AS, fct1)/as_reduction(AS, fct2)
+    
+    result = RxyzQ(0)
+    change = 0
+    for a in fct1.monomials():
+        degrees_zi = [a.degree(z[i]) for i in range(n)]
+        d_div = [a.degree(z[i])//p for i in range(n)]
+        if d_div != n*[0]:
+            change = 1
+        d_rem = [a.degree(z[i])%p for i in range(n)]
+        monomial = fct1.monomial_coefficient(a)*x^(a.degree(x))*y^(a.degree(y))*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i] + ff[i])^(d_div[i]) for i in range(n))
+        result += RxyzQ(monomial)        
+        
+    if change == 0:
+        return RxyzQ(result)
+    else:
+        return as_reduction(AS, RxyzQ(result))

sage/as_covers/group_action_matrices.sage

@@ -1,4 +1,34 @@
-def group_action_matrices(C_AS):
+def group_action_matrices(space, list_of_group_elements, basis):
+    n = len(list_of_group_elements)
+    d = len(space)
+    A = [matrix(F, d, d) for i in range(n)]
+    for i, g in enumerate(list_of_group_elements):
+        for j, omega in enumerate(space):
+            omega1 = omega.group_action(g)
+            v1 = omega1.coordinates(basis = basis)
+            A[i][:, j] = vector(v1)
+    return A
+
+def group_action_matrices_holo(AS):
+    n = AS.height
+    generators = []
+    for i in range(n):
+        ei = n*[0]
+        ei[i] = 1
+        generators += [ei]
+    return group_action_matrices(AS.holomorphic_differentials_basis(), generators, basis = AS.holomorphic_differentials_basis())
+    
+def group_action_matrices_dR(AS, threshold=8):
+    n = AS.height
+    generators = []
+    for i in range(n):
+        ei = n*[0]
+        ei[i] = 1
+        generators += [ei]
+    basis = [AS.holomorphic_differentials_basis(), AS.cohomology_of_structure_sheaf_basis(), AS.de_rham_basis(threshold=threshold)]
+    return group_action_matrices(basis[2], generators, basis = basis)
+    
+def group_action_matrices_old(C_AS):
     F = C_AS.base_ring
     n = C_AS.height
     holo = C_AS.holomorphic_differentials_basis()

sage/as_covers/tests/group_action_matrices_test.sage

@@ -10,7 +10,7 @@ f1 = superelliptic_function(C_super, x^2*y)
 f2 = superelliptic_function(C_super, x^3)
 AS = as_cover(C_super, [f1, f2], prec=1000)
 
-A, B = group_action_matrices(AS)
+A, B = group_action_matrices_holo(AS)
 n = A.dimensions()[0]
 print(A*B == B*A)
 print(A^p == identity_matrix(n))

sage/init.sage

@@ -5,6 +5,7 @@ load('superelliptic/superelliptic_cech_class.sage')
 load('as_covers/as_cover_class.sage')
 load('as_covers/as_function_class.sage')
 load('as_covers/as_form_class.sage')
+load('as_covers/as_cech_class.sage')
 load('as_covers/as_reduction.sage')
 load('as_covers/as_auxilliary.sage')
 load('as_covers/dual_element.sage')
@@ -17,6 +18,6 @@ load('auxilliaries/linear_combination_polynomials.sage')
 ##############
 ##############
 load('drafty/lift_to_de_rham.sage')
-load('drafty/draft3.sage')
+load('drafty/draft5.sage')
 load('drafty/pole_numbers.sage')
 #load('drafty/draft4.sage')

sage/tests.sage

@@ -1,16 +1,16 @@
 #print("as_cover_test:")
 #load('as_covers/tests/as_cover_test.sage')
 #print("group_action_matrices_test:")
-#load('as_covers/tests/group_action_matrices_test.sage')
+load('as_covers/tests/group_action_matrices_test.sage')
 #print("dual_element_test:")
 #load('as_covers/tests/dual_element_test.sage')
-print("ith_component_test:")
-load('as_covers/tests/ith_component_test.sage')
+#print("ith_component_test:")
+#load('as_covers/tests/ith_component_test.sage')
 #print("ith ramification group test:")
 #load('as_covers/tests/ith_ramification_gp_test.sage')
 #print("uniformizer test:")
 #load('as_covers/tests/uniformizer_test.sage')
 #print("ramification jumps test:")
 #load('as_covers/tests/ramification_jumps_test.sage')
-print("diffn_test:")
-load('as_covers/tests/diffn_test.sage')
+#print("diffn_test:")
+#load('as_covers/tests/diffn_test.sage')