Poprawiony i sprawdzony dy_w oraz autom.

sage/drafty/draft.sage

@@ -13,4 +13,4 @@ C = superelliptic(f, m)
 #converted = (C2.x)^4 - (C2.x)^2
 #print(convert_super_fct_into_AS(converted))
 b = C.crystalline_cohomology_basis()
-print(autom(b[0]).coordinates(basis = b))
+p#rint(autom(b[0]).coordinates(basis = b))

sage/drafty/superelliptic_drw.sage

@@ -43,7 +43,7 @@ class superelliptic_witt:
             if other < 0:
                 return (-other)*(-self)
         if other in QQ:
-            other_integer = Integers(p^2)(other)^(-1)
+            other_integer = Integers(p^2)(other)
             return other_integer*self
     
     def __mul__(self, other):
@@ -67,7 +67,9 @@ class superelliptic_witt:
     def __eq__(self, other):
         return self.t == other.t and self.f == other.f
     
-    def diffn(self):
+    def diffn(self, dy_w = 0):
+        if dy_w == 0:
+            dy_w = self.curve.dy_w()
         C = self.curve
         t = self.t
         f = self.f
@@ -86,9 +88,6 @@ class superelliptic_witt:
             # d([N/D] + V(f)) = [1/D]*d([N]) - [N]*[D^(-2)]*d([D]) + dV(f)
             return ((den_t_fct)^(-1)).teichmuller()*num_t_fct.teichmuller().diffn() - ((den_t_fct)^(-2)).teichmuller()*num_t_fct.teichmuller()*den_t_fct.teichmuller().diffn() + superelliptic_drw_form(0*C.x, 0*C.dx, f)
         t_polynomial = Rxy(t_polynomial)
-        fC = superelliptic_function(C, fC)
-        fC = fC.teichmuller()
-        dy_w = 1/2* ((C.y)^(-1)).teichmuller()*auxilliary_derivative(fC)
         M = t_polynomial.monomials()[0]
         a = t_polynomial.monomial_coefficient(M)
         #[P] = [aM] + Q, where Q = ([P] - [aM])
@@ -167,7 +166,7 @@ class superelliptic_drw_form:
             H = (self.h2 - other.h2).pth_root()
         except:
             return False
-        eq2 = (self.omega - other.omega).cartier() - H.diffn()
+        eq2 = ((self.omega - other.omega).cartier() - H.diffn()) == 0*self.curve.dx
         if eq1 and eq2:
             return True
         return False
@@ -205,7 +204,7 @@ class superelliptic_drw_form:
             if other < 0:
                 return (-other)*(-self)
         if other in QQ:
-            other_integer = Integers(p^2)(other)^(-1)
+            other_integer = Integers(p^2)(other)
             return other_integer*self
         t = other.t
         f = other.f
@@ -334,9 +333,6 @@ class superelliptic_drw_cech:
             aux -= coord_lifted[i]*a
         print('aux before reduce', aux)
         #aux = aux.reduce() # Now aux = p*cech class.
-        h02 = aux.omega0.h2
-        h82 = aux.omega8.h2
-        aux -= superelliptic_drw_cech(h02.verschiebung().diffn(), (h02 - h82).verschiebung())
         # 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())
@@ -346,6 +342,12 @@ class superelliptic_drw_cech:
         coordinates = [ (coord_lifted[i] + p*coord_aux_divided_by_p[i])%p^2 for i in range(2*g)]
         return coordinates
     
+    def is_regular(self):
+        print(self.omega0.r().is_regular_on_U0(), self.omega8.r().is_regular_on_Uinfty(), self.omega0.frobenius().is_regular_on_U0(),     self.omega8.frobenius().is_regular_on_Uinfty())
+        eq1 = self.omega0.r().is_regular_on_U0() and self.omega8.r().is_regular_on_Uinfty()
+        eq2 = self.omega0.frobenius().is_regular_on_U0() and self.omega8.frobenius().is_regular_on_Uinfty()
+        return eq1 and eq2
+        
 
 def de_rham_witt_lift(cech_class, prec = 50):
     C = cech_class.curve
@@ -354,7 +356,6 @@ def de_rham_witt_lift(cech_class, prec = 50):
     omega8 = cech_class.omega8
     fct = cech_class.f
     omega0_regular = regular_form(omega0) #Present omega0 in the form P dx + Q dy
-    print(omega0_regular)
     omega0_lift = omega0_regular[0].teichmuller()*(C.x.teichmuller().diffn()) + omega0_regular[1].teichmuller()*(C.y.teichmuller().diffn())
     #Now the obvious lift of omega0 = P dx + Q dy to de Rham-Witt is [P] d[x] + [Q] d[y]
     omega8_regular = regular_form(second_patch(omega8)) # The same for omega8.
@@ -362,7 +363,6 @@ def de_rham_witt_lift(cech_class, prec = 50):
     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())
-    print('omega8_lift.frobenius().expansion_at_infty()', omega8_lift.frobenius().expansion_at_infty())
     aux = omega0_lift - omega8_lift - fct.teichmuller().diffn() # now aux is of the form (V(smth) + dV(smth), V(smth))
     if aux.h1.function != 0:
         raise ValueError('Something went wrong - aux is not of the form (V(smth) + dV(smth), V(smth)).')
@@ -396,8 +396,19 @@ def autom(self):
         result = superelliptic_witt(autom(self.t), autom(self.f))
         return result
     if isinstance(self, superelliptic_drw_form):
-        result = superelliptic_drw_form(autom(self.h1), autom(self.omega), autom(self.h2))
+        result = superelliptic_drw_form(0*C.x, autom(self.omega), autom(self.h2))
+        result += autom(self.h1).teichmuller()*(C.x + C.one).teichmuller().diffn()
         return result
     if isinstance(self, superelliptic_drw_cech):
         result = superelliptic_drw_cech(autom(self.omega0), autom(self.f))
         return result
+    
+    
+def dy_w(C):
+    '''Return d[y].'''
+    fC = C.polynomial
+    fC = superelliptic_function(C, fC)
+    fC = fC.teichmuller()
+    dy_w = 1/2* ((C.y)^(-1)).teichmuller()*auxilliary_derivative(fC)
+    return dy_w
+superelliptic.dy_w = dy_w

sage/superelliptic/decomposition_into_g0_g8.sage

@@ -57,3 +57,10 @@ def decomposition_omega0_omega8(omega, prec=50):
     #aux_fct = (g0.form)*y
     else:
         raise ValueError("Something went wrong for "+str(omega) +". Result would be "+str(g0)+ " and " + str(g8))
+        
+        
+def decomposition_g0_g8_pth_power(fct):
+    '''Decompose fct as g0 - g8 + A^p, if possible. Output: (g0, g8, A).'''
+    coor = fct.coordinates()
+    C = fct.curve
+    return 0