baza crys dziala; poprawki w coordinates dr; wspolrzednie cris prawie dzialaja

sage/as_covers/tests/cartier_test.sage

@@ -0,0 +1,11 @@
+p = 5
+m = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^3 + x + 1
+C = superelliptic(f, m)
+g = f(x^p - x)
+C1 = superelliptic(g, m)
+ff = superelliptic_function(C, x)
+AS = as_cover(C, [ff])
+print(C1.cartier_matrix().rank() == AS.cartier_matrix().rank())

sage/drafty/draft.sage

@@ -15,8 +15,14 @@ C = superelliptic(f, m)
 #b = C.crystalline_cohomology_basis()
 #print(autom(b[0]).coordinates(basis = b))
 #eta1 = (dy + dV(2xy) + V(x^5 \, dy), V(y/x))
-eta1 = superelliptic_drw_cech(C.y.teichmuller().diffn() + (2*C.x*C.y).verschiebung().diffn() + (C.x^5*C.y.diffn()).verschiebung(), (C.y/C.x).verschiebung())
+#eta1 = superelliptic_drw_cech(C.y.teichmuller().diffn() + (2*C.x*C.y).verschiebung().diffn() + (C.x^5*C.y.diffn()).verschiebung(), (C.y/C.x).verschiebung())
 #eta2 =  (  x \, dy + 3 x^3 \, dy + dV((2x^4 + 2x^2 + 2) y) + V( (x^4 + x^2 + 1) dy), -[y/x])
-eta2 = superelliptic_drw_cech(C.x.teichmuller()*(C.y.teichmuller()).diffn() + ((2*C.x^4 + 2*C.x^2 + 2*C.one) * C.y).verschiebung().diffn(), - (C.y/C.x).teichmuller())
+#eta2 = superelliptic_drw_cech(C.x.teichmuller()*(C.y.teichmuller()).diffn() + ((2*C.x^4 + 2*C.x^2 + 2*C.one) * C.y).verschiebung().diffn(), - (C.y/C.x).teichmuller())
-aux = de_rham_witt_lift(C.de_rham_basis()[1])
+#omega8_lift0, compare = de_rham_witt_lift(C.de_rham_basis()[1])
-print(aux)
+#omega8_lift = -(C.x^(-3)).teichmuller()*C.y.teichmuller().diffn() + 2*C.y.teichmuller()*(C.x^(-4)).teichmuller()*C.x.teichmuller().diffn()
+#eta2 = de_rham_witt_lift(C.de_rham_basis()[1])
+#b = autom(eta2)
+#print(autom(C.crystalline_cohomology_basis()[1]).coordinates())
+
+
+

sage/init.sage

@@ -18,6 +18,7 @@ load('superelliptic_drw/decomposition_into_g0_g8.sage')
 load('superelliptic_drw/superelliptic_witt.sage')
 load('superelliptic_drw/superelliptic_drw_form.sage')
 load('superelliptic_drw/superelliptic_drw_cech.sage')
+load('superelliptic_drw/regular_form.sage')
 load('superelliptic_drw/de_rham_witt_lift.sage')
 load('superelliptic_drw/automorphism.sage')
 load('auxilliaries/reverse.sage')
@@ -26,8 +27,6 @@ load('auxilliaries/linear_combination_polynomials.sage')
 ##############
 ##############
 load('drafty/convert_superelliptic_into_AS.sage')
-load('drafty/regular_on_U0.sage')
-load('drafty/lift_to_de_rham.sage')
 load('drafty/draft.sage')
 #load('drafty/draft_klein_covers.sage')
 #load('drafty/2gpcovers.sage')

sage/superelliptic/superelliptic_cech_class.sage

@@ -46,52 +46,30 @@ class superelliptic_cech:
         Fx = FractionField(Rx)
         FxRy.<y> = PolynomialRing(Fx)
         g = C.genus()
-        degrees_holo = C.degrees_holomorphic_differentials()
-        degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
-        degrees0 = C.degrees_de_rham0()
-        degrees0_inv = {b:a for a, b in degrees0.items()}
-        degrees1 = C.degrees_de_rham1()
-        degrees1_inv = {b:a for a, b in degrees1.items()}
         basis = C.de_rham_basis()
         
         omega = self.omega0
         fct = self.f
         
+        if fct.function == Rx(0) and omega.form == Rx(0):
+            return vector((2*g)*[0])
+        
         if fct.function == Rx(0) and omega.form != Rx(0):
-            for j in range(1, m):
+            result = list(omega.coordinates()) + g*[0]
-                omega_j = Fx(omega.jth_component(j))
+            result = vector([F(a) for a in result])
-                if omega_j != Fx(0):
+            return result
-                    d = degree_of_rational_fctn(omega_j, F)
-                    index = degrees_holo_inv[(d, j)]
-                    a = coeff_of_rational_fctn(omega_j, F)
-                    a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), F)
-                    elt = self - (a/a1)*basis[index]
-                    return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, 2*g)])
         
-        for j in range(1, m):
+        coord = fct.coordinates()
-            fct_j = Fx(fct.jth_component(j))
+        coord = g*[0] + list(coord)
-            if (fct_j != Rx(0)):
+        coord = vector([F(a) for a in coord])
-                d = degree_of_rational_fctn(fct_j, F)
+        aux = self
-            
+        for i in range(g, 2*g):
-                if (d, j) in degrees1.values():
+            aux -= coord[i]*basis[i]
-                    index = degrees1_inv[(d, j)]
+        aux_f = decomposition_g0_g8(aux.f)[0]
-                    a = coeff_of_rational_fctn(fct_j, F)
+        aux.omega0 -= aux_f.diffn()
-                    elt = self - (a/m)*basis[index]
+        aux.f = 0*C.x
-                    return elt.coordinates() + a/m*vector([F(i == index) for i in range(0, 2*g)])
+        aux.omega8 = aux.omega0
-                
+        return coord + aux.coordinates()
-                if d<0:
-                    a = coeff_of_rational_fctn(fct_j, F)
-                    h = superelliptic_function(C, FxRy(a*y^j*x^d))
-                    elt = superelliptic_cech(C, self.omega0, self.f - h)
-                    return elt.coordinates()
-            
-                if (fct_j != Rx(0)):
-                    G = superelliptic_function(C, y^j*x^d)
-                    a = coeff_of_rational_fctn(fct_j, F)
-                    elt =self - a*superelliptic_cech(C, diffn(G), G)
-                    return elt.coordinates()
-
-        return vector(2*g*[0])
     
     def is_cocycle(self):
         w0 = self.omega0

sage/superelliptic/superelliptic_class.sage

@@ -196,9 +196,9 @@ class superelliptic:
         Fxy = FractionField(Rxy)
         basis = []
         for j in range(1, m):
-                for i in range(1, r):
+            for i in range(1, r):
-                    if (r*j - m*i >= delta):
+                if (r*(m-j) - m*i >= delta): 
-                        basis += [superelliptic_function(self, Fxy(m*y^(j)/x^i))]
+                    basis += [superelliptic_function(self, Fxy(m*y^(m-j)/x^i))]
         return basis
 
 #Auxilliary. Given a superelliptic curve C : y^m = f(x) and a polynomial g(x, y)

sage/superelliptic/superelliptic_form_class.sage

@@ -7,6 +7,11 @@ class superelliptic_form:
         self.form = g
         self.curve = C
     
+    def __eq__(self, other):
+        if self.reduce().form == other.reduce().form:
+            return True
+        return False
+    
     def __add__(self, other):
         C = self.curve
         g1 = self.form
@@ -37,9 +42,6 @@ class superelliptic_form:
         omega = self.form
         return superelliptic_form(C, constant*omega)
     
-    def __eq__(self, other):
-        return self.form == other.form
-    
     def cartier(self):
         '''Computes Cartier operator on the form. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
         M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus C(h(x)/y^j dx) = 1/y^(p^(r-1)*j) C(h(x) f(x)^(M*j) dx).'''
@@ -52,7 +54,7 @@ class superelliptic_form:
         Fx = FractionField(Rx)
         FxRy.<y> = PolynomialRing(Fx)
         Fxy = FractionField(FxRy)
-        result = superelliptic_form(C, FxRy(0))
+        result = 0*C.dx
         mult_order = Integers(m)(p).multiplicative_order()
         M = Integer((p^(mult_order)-1)/m)
         
@@ -111,8 +113,8 @@ class superelliptic_form:
         Rx.<x> = PolynomialRing(F)
         for j in range(0, m):
             if self.jth_component(j) not in Rx:
-                return 0
+                return False
-        return 1
+        return True
     
     def is_regular_on_Uinfty(self):
         C = self.curve
@@ -128,8 +130,8 @@ class superelliptic_form:
             A = self.jth_component(j)
             d = degree_of_rational_fctn(A, F)
             if(-d*M + j*R -(M+1)<0):
-                return 0
+                return False
-        return 1
+        return True
 
     def expansion_at_infty(self, place = 0, prec=10):
         g = self.form
@@ -167,3 +169,37 @@ class superelliptic_form:
         Fxy = FractionField(Rxy)
         fct = reduction(C, Fxy(y^m*fct))
         return superelliptic_form(C, fct/y^m)
+    
+    def int(self):
+        '''Computes an "integral" of a form dg. Idea: y^m = f(x) -> y^(p^r - 1) = f(x)^M, where r = ord_p(m),
+        M = (p^r - 1)/m. Thus h(x)/y^j dx = h(x) f(x)^(M*j)/y^(p^r * j) dx. Thus int(h(x)/y^j dx) = 1/y^(p^(r-1)*j) int(h(x) f(x)^(M*j) dx).'''
+        C = self.curve
+        m = C.exponent
+        p = C.characteristic
+        f = C.polynomial
+        F = C.base_ring
+        Rx.<x> = PolynomialRing(F)
+        Fx = FractionField(Rx)
+        FxRy.<y> = PolynomialRing(Fx)
+        Fxy = FractionField(FxRy)
+        result = 0*C.x
+        mult_order = Integers(m)(p).multiplicative_order()
+        M = Integer((p^(mult_order)-1)/m)
+        
+        for j in range(0, m):
+            fct_j = self.jth_component(j)
+            h = Fx(fct_j*f^(M*j))
+            h_denom = h.denominator()
+            h *= (h_denom)^(p)
+            h = Rx(h)
+            j1 = (p^(mult_order)*j)%m
+            B = floor(p^(mult_order)*j/m)
+            result += superelliptic_function(C, h.integral()/(f^(B)*y^(j1)*h_denom^p))
+        return result   
+    
+    def inv_cartier(omega):
+        '''If omega is regular, return form eta such that Cartier(eta) = omega'''
+        omega_regular = omega.regular_form()
+        C = omega.curve
+        p = C.characteristic
+        return (omega_regular.dx)^p*C.x^(p-1)*C.dx + (omega_regular.dy)^p*C.y^(p-1)*C.y.diffn()

sage/superelliptic/tests/a_number_test.sage

@@ -0,0 +1,5 @@
+F = GF(67)
+P.<x>= PolynomialRing(F)
+X = HyperellipticCurve(x^7 + x^3 + x)
+C = superelliptic(x^7 + x^3 + x, 2)
+print(X.a_number() == C.a_number())

sage/superelliptic/tests/form_coordinates_test.sage

@@ -0,0 +1,12 @@
+p = 7
+m = 4
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^5 + x
+C = superelliptic(f, m)
+bbb = C.holomorphic_differentials_basis()
+v = [GF(p).random_element() for _ in range(C.genus())]
+aaa = 0*C.dx
+for i in range(C.genus()):
+    aaa += v[i]*bbb[i]
+print(vector(aaa.coordinates()) == vector(v))

sage/superelliptic/tests/p_rank_test.sage

@@ -0,0 +1,6 @@
+print("Nie działa!")
+F = GF(67)
+P.<x>= PolynomialRing(F)
+X = HyperellipticCurve(x^7 + x^3 + x)
+C = superelliptic(x^7 + x^3 + x, 2)
+print(X.p_rank() == C.p_rank())

sage/superelliptic/tests/pth_root_test.sage

@@ -0,0 +1,11 @@
+p = 3
+m = 4
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^5 + x
+C = superelliptic(f, m)
+g = (C.x)^5 * (C.y)^2 + 2*(C.x)^2 * (C.y)^3
+g = g^p
+print(g.pth_root()==(C.x)^5 * (C.y)^2 + 2*(C.x)^2 * (C.y)^3)
+g = C.x
+print(g.pth_root())

sage/superelliptic_drw/automorphism.sage

@@ -11,6 +11,9 @@ def autom(self):
     if isinstance(self, superelliptic_form):
         result = superelliptic_form(C, Fxy(self.form).subs({x:x+1, y:y}))
         return result
+    if isinstance(self, superelliptic_cech):
+        result = superelliptic_cech(C, autom(self.omega0), autom(self.f))
+        return result
     if isinstance(self, superelliptic_witt):
         result = superelliptic_witt(autom(self.t), autom(self.f))
         return result

sage/superelliptic_drw/de_rham_witt_lift.sage

@@ -1,31 +1,50 @@
+def de_rham_witt_lift_form0(omega):
+    C = omega.curve
+    omega_regular = omega.regular_form() #Present omega0 in the form P dx + Q dy
+    #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
+    return omega_regular.dx.teichmuller()*(C.x.teichmuller().diffn()) + omega_regular.dy.teichmuller()*(C.y.teichmuller().diffn())
+
+def de_rham_witt_lift_form8(omega):
+    C = omega.curve
+    g = C.genus()
+    omega_regular = second_patch(omega).regular_form()
+    omega_regular = (second_patch(omega_regular.dx), second_patch(omega_regular.dy))
+    u = (C.x)^(-1)
+    v = (C.y)/(C.x)^(g+1)
+    omega_lift = omega_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega_regular[1].teichmuller()*(v.teichmuller().diffn())
+    return omega_lift
+    
 def de_rham_witt_lift(cech_class, prec = 50):
     C = cech_class.curve
     g = C.genus()
     omega0 = cech_class.omega0
     omega8 = cech_class.omega8
     fct = cech_class.f
-    omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
+    omega0_lift = de_rham_witt_lift_form0(omega0)
-    print('omega0_regular', omega0_regular)
+    omega8_lift = de_rham_witt_lift_form8(omega8)
-    omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
+    print('omega0_lift, omega8_lift', omega0_lift, omega8_lift)
-    #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
-    print('omega8', omega8, 'second_patch(omega8)', second_patch(omega8))
-    omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
-    print('omega8_regular 1', omega8_regular)
-    omega8_regular = (second_patch(omega8_regular[0]), second_patch(omega8_regular[1]))
-    print('omega8_regular 2', omega8_regular)
-    u = (C.x)^(-1)
-    v = (C.y)/(C.x)^(g+1)
-    omega8_lift = omega8_regular[0].teichmuller()*(u.teichmuller().diffn()) + omega8_regular[1].teichmuller()*(v.teichmuller().diffn())
     aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
-    return aux
+    #return aux
     if aux.h1.function != 0:
         raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
     decom_aux_h2 = decomposition_g0_g8(aux.h2, prec=prec) #decompose dV(smth) in aux as smth regular on U0 - smth regular on U8.
     aux_h2 = decom_aux_h2[0]
     aux_f = decom_aux_h2[2]
     aux_omega0 = decomposition_omega0_omega8(aux.omega, prec=prec)[0]
-    result = superelliptic_drw_cech(omega0_lift - aux_h2.verschiebung().diffn() - aux_omega0.verschiebung(), fct.teichmuller() + aux_f.verschiebung())
+    result = superelliptic_drw_cech(omega0_lift - aux_omega0.verschiebung(), fct.teichmuller() + aux_h2.verschiebung() + aux_f.verschiebung())
-    return result.reduce()
+    compare = omega8_lift-decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
+    print("result.omega8 == compare", result.omega8 == compare)
+    print("result.omega8 - compare", result.omega8 - compare)
+    
+    #print('test:', omega0_lift - omega8_lift - fct.teichmuller().diffn() == decom_aux_h2[0].verschiebung().diffn() - decom_aux_h2[1].verschiebung().diffn() + decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
+    #print('test 1:', omega0_lift - decom_aux_h2[0].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung() - fct.teichmuller().diffn() == omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung())
+    #A = omega0_lift - decomposition_omega0_omega8(aux.omega, prec=prec)[0].verschiebung()
+    #B = decom_aux_h2[0].verschiebung() + fct.teichmuller()
+    #C = omega8_lift - decom_aux_h2[1].verschiebung().diffn() - decomposition_omega0_omega8(aux.omega, prec=prec)[1].verschiebung()
+    #print('test 2:', A - B.diffn() == C)
+    #print('test 3:', result.omega0 == A, result.f == B, result.omega8 == C)
+    #print(result.omega8, '\n \n', compare, '\n \n', aux_f, '\n \n')
+    return result#.reduce()
 
 def crystalline_cohomology_basis(self, prec = 50):
     result = []

sage/superelliptic_drw/decomposition_into_g0_g8.sage

@@ -3,7 +3,7 @@ def decomposition_g0_g8(fct, prec = 50):
     and f is combination of basis of H^1(X, OX). Output is (g0, g8, f).'''
     C = fct.curve
     g = C.genus()
-    coord = fct.coordinates()
+    coord = fct.coordinates(prec=prec)
     nontrivial_part = 0*C.x
     for i, a in enumerate(C.cohomology_of_structure_sheaf_basis()):
         nontrivial_part += coord[i]*a
@@ -13,8 +13,9 @@ def decomposition_g0_g8(fct, prec = 50):
     fct = Fxy(fct.function)
     num = fct.numerator()
     den = fct.denominator()
+    integral_part, num = num.quo_rem(den)
     aux_den = superelliptic_function(C, Rxy(den))
-    g0 = superelliptic_function(C, 0)
+    g0 = superelliptic_function(C, integral_part)
     g8 = superelliptic_function(C, 0)
     for monomial in num.monomials():
         aux = superelliptic_function(C, monomial)
@@ -31,7 +32,7 @@ def decomposition_omega0_omega8(omega, prec=50):
     F = C.base_ring
     delta = C.nb_of_pts_at_infty
     m = C.exponent
-    if sum(omega.residue(place = i, prec = 50) for i in range(delta)) != 0:
+    if sum(omega.residue(place = i, prec = prec) for i in range(delta)) != 0:
         raise ValueError(str(omega) + " has non zero residue!")
     Fxy, Rxy, x, y = C.fct_field
     Rx.<x> = PolynomialRing(F)

sage/superelliptic_drw/second_patch.sage

@@ -23,6 +23,13 @@ def second_patch(argument):
         fct1 = Fxy(fct.subs({x : 1/x, y : y/x^(g+1)}))
         fct1 *= -x^(-2)
         return superelliptic_form(C1, fct1)
+    if isinstance(argument, superelliptic_drw_form):
+        h1 = argument.h1
+        omega = argument.omega
+        h2 = argument.h2
+        C = h1.curve
+        return superelliptic_drw_form(-second_patch(h1)*(C.x)^(-2), second_patch(omega), second_patch(h2))
+    
     
 def lift_form_to_drw(omega):
     A, B = regular_form(omega)

sage/superelliptic_drw/superelliptic_drw_cech.sage

@@ -56,6 +56,42 @@ class superelliptic_drw_cech:
         C = self.curve
         return superelliptic_cech(C, omega0.h1*C.dx, f.t)
     
+    def div_by_p(self):
+        '''Given a regular cocycle of the form (V(omega) + dV(h), [f] + V(t), ...), where [f] = 0 in H^1(X, OX),
+        find de Rham cocycle (xi0, f, xi8) such that (V(omega) + dV(h), [f] + V(t), ...) = p*(xi0, f, xi8).'''
+        C = self.curve
+        aux = self
+        Fxy, Rxy, x, y = C.fct_field
+        aux_f_t_0 = decomposition_g0_g8(aux.f.t, prec=50)[0]
+        aux.f.t = 0*C.x
+        aux.omega0 -= aux_f_t_0.teichmuller().diffn()
+        aux.omega8 = aux.omega0 - aux.f.diffn()
+        #
+        omega = aux.omega0.omega
+        omega1 = omega.cartier().cartier()
+        omega1 = omega1.inv_cartier().inv_cartier()
+        fct = (omega.cartier() - omega1.cartier()).int()
+        aux.omega0.h2 += fct^p
+        aux.omega0.omega = omega1
+        if aux.omega0.h2.function in Rxy:
+            aux.f -= aux.omega0.h2.verschiebung()
+            aux.omega0.h2 = 0*C.x
+            if aux.omega8.h2.expansion_at_infty().valuation() >= 0:
+                aux.f += aux.omega8.h2.verschiebung()
+                aux.omega8.h2 = 0*C.x
+                print('aux', aux)
+                # Now aux should be of the form (V(omega1), V(f), V(omega2))
+                # Thus aux = p*(Cartier(omega1), p-th_root(f), Cartier(omega2))
+                aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
+                print('aux_divided_by_p', aux_divided_by_p)
+                print('is regular', aux_divided_by_p.omega0.is_regular_on_U0(), aux_divided_by_p.omega8.is_regular_on_Uinfty())
+                print('aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier()', aux.omega0.omega.cartier() - aux.f.f.pth_root().diffn() == aux.omega8.omega.cartier())
+                return aux_divided_by_p
+            else:
+                raise ValueError("aux.omega8.h2.expansion_at_infty().valuation() < 0:", aux.omega8.h2.expansion_at_infty())
+        else:
+            raise ValueError("aux.omega0.h2.function not in Rxy:", aux.omega0.h2.function)
+        
     def coordinates(self, basis = 0):
         C = self.curve
         g = C.genus()
@@ -67,12 +103,7 @@ class superelliptic_drw_cech:
         aux = self
         for i, a in enumerate(basis):
             aux -= coord_lifted[i]*a
-        print('aux before reduce', aux)
+        aux_divided_by_p = aux.div_by_p()
-        #aux = aux.reduce() # Now aux = p*cech class.
-        # Now aux should be of the form (V(smth), V(smth), V(smth))
-        print('aux V(smth)', aux)
-        aux_divided_by_p = superelliptic_cech(C, aux.omega0.omega.cartier(), aux.f.f.pth_root())
-        print('aux.omega0.omega.cartier()', aux.omega0.omega.cartier())
         coord_aux_divided_by_p = aux_divided_by_p.coordinates()
         coord_aux_divided_by_p = [ZZ(a) for a in coord_aux_divided_by_p]
         coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]

sage/superelliptic_drw/superelliptic_drw_form.sage

@@ -17,7 +17,7 @@ class superelliptic_drw_form:
             H = (self.h2 - other.h2).pth_root()
         except:
             return False
-        eq2 = ((self.omega - other.omega).cartier() - H.diffn()) == 0*self.curve.dx
+        eq2 = ((other.omega - self.omega).cartier() - H.diffn()) == 0*self.curve.dx
         if eq1 and eq2:
             return True
         return False
@@ -30,7 +30,7 @@ class superelliptic_drw_form:
         result = ""
         if h1.function != 0:
             result += "[" + str(h1) + "] d[x]"
-        if h1.function !=0 and omega.form != 0:
+        if (h1.function !=0 and omega.form != 0) or (h2.function !=0 and omega.form != 0):
             result += " + "
         if omega.form != 0:
             result += "V(" + str(omega) + ")"

sage/superelliptic_drw/tests/decomposition_into_g0_g8_tests.sage

@@ -0,0 +1,11 @@
+p = 3
+m = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^3 - x
+C = superelliptic(f, m)
+Rxy.<x, y> = PolynomialRing(F, 2)
+omega = (((2*C.x^18 + 2*C.x^16 + 2*C.x^14 + 2*C.x^10 + 2*C.x^8 + 2*C.x^4 + 2*C.x^2 + 2*C.one)/(C.x^13 + C.x^11 + C.x^9))*C.y) * C.dx
+print(decomposition_omega0_omega8(aux.omega)[0] - decomposition_omega0_omega8(aux.omega)[1] == omega and decomposition_omega0_omega8(aux.omega)[0].is_regular_on_U0() and decomposition_omega0_omega8(aux.omega)[1].is_regular_on_Uinfty())
+h = ((C.x^10 + C.x^8 + C.x^6 + 2*C.x^4 + 2*C.x^2 + 2*C.one)/C.x^6)*C.y
+print(decomposition_g0_g8(h)[0] - decomposition_g0_g8(h)[1] + decomposition_g0_g8(h)[2] == h and decomposition_g0_g8(h)[0].function in Rxy and decomposition_g0_g8(h)[1].expansion_at_infty().valuation() >= 0)

sage/superelliptic_drw/tests/superelliptic_drw_tests.sage

@@ -0,0 +1,10 @@
+p = 3
+m = 2
+F = GF(p)
+Rx.<x> = PolynomialRing(F)
+f = x^3 - x
+C = superelliptic(f, m)
+print(auxilliary_derivative((C.x^3 - C.x).teichmuller()))
+print('Result should be: [2] d[x] + V((x^8) dx) + dV([2*x^7 + x^5])')
+print(2*(C.y).teichmuller() * (C.y).teichmuller().diffn() == (C.x^3 - C.x).teichmuller().diffn())
+print(C.y.teichmuller().diffn().frobenius() == (C.y)^2 * C.y.diffn()) #F(d[y]) = y^2*dy

sage/tests.sage

@@ -25,3 +25,5 @@ load('superelliptic/tests/a_number_test.sage')
 #load('as_covers/tests/diffn_test.sage')
 #print("Cartier test:")
 #load('as_covers/tests/cartier_test.sage')
+#print("Decomposition into g0, g8/ omega0, omega8 test:")
+#load('superelliptic_drw/tests/decomposition_into_g0_g8_tests.sage')