new nth_root procedure for power series

as_covers/as_form_class.sage

@@ -97,7 +97,6 @@ class as_form:
         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.
-        print([denominator(omega.form) for omega in basis])
         denom = LCM([denominator(omega.form) for omega in basis])
         basis = [denom*omega for omega in basis]
         self_with_no_denominator = denom*self
@@ -194,12 +193,12 @@ def artin_schreier_transform(power_series, prec = 10):
         power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))
     jump = max(-(power_series.valuation()), 0)
     try:
-        T = ((power_series)^(-1)).nth_root(jump) #T is defined by power_series = 1/T^m
+        T = nth_root2((power_series)^(-1), jump, prec=prec) #T is defined by power_series = 1/T^m
     except:
         print("no ", str(jump), "-th root; divide by", power_series.list()[0])
         return (jump, power_series.list()[0])
     T_rev = new_reverse(T, prec = prec)
-    t_old = T_rev(t^p/(1 - t^((p-1)*jump)).nth_root(jump))
+    t_old = T_rev(t^p/nth_root2(1 - t^((p-1)*jump), jump, prec=prec))
     z = 1/t^(jump) + Rt(correction)(t = t_old)
     return(jump, correction, t_old, z)
 

as_covers/group_action_matrices.sage

@@ -27,7 +27,7 @@ def group_action_matrices_dR(AS, threshold=8):
         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)]
+    basis = [AS.holomorphic_differentials_basis(threshold = threshold), AS.cohomology_of_structure_sheaf_basis(threshold = threshold), AS.de_rham_basis(threshold=threshold)]
     return group_action_matrices(basis[2], generators, basis = basis)
     
 def group_action_matrices_old(C_AS):

as_covers/tests/uniformizer_test.sage

@@ -9,4 +9,4 @@ Rxy.<x, y> = PolynomialRing(F, 2)
 f1 = superelliptic_function(C_super, x^7)
 f2 = superelliptic_function(C_super, x^4)
 AS = as_cover(C_super, [f1, f2], prec=1000)
-print(AS.unifomizer().valuation() == 1)
+print(AS.uniformizer().valuation() == 1)

auxilliaries/hensel.sage

@@ -16,4 +16,16 @@ def naive_hensel(fct, F, start = 1, prec=10):
     while(i < prec):
         w0 = w0 - fct(W = w0)/alpha + O(t^(prec))
         i += 1
-    return w0
+    return w0
+
+def nth_root2(fct, n, prec=10):
+    '''Given power series in F((t)), find its n-th root up to precision prec.'''
+    F= parent(fct).base_ring()
+    Rt.<t> = LaurentSeriesRing(F, default_prec=prec)
+    RW.<W> = PolynomialRing(Rt)
+    v = fct.valuation()
+    fct1 = Rt(fct*t^(-v))
+    a0 = fct1[0]
+    if v%n != 0:
+        raise ValueError('The valuation of the power series is not divisible by n.')
+    return t^(v//n)*naive_hensel(W^n - fct1, F, start = a0.nth_root(n), prec=prec)

drafty/draft2.sage

@@ -1,18 +1,15 @@
-def reduction(g):
-    F = g.parent().base()
-    x, y = g.parent().gens()
-    Rxy.<x, y> = PolynomialRing(F, 2)
-    Fxy = FractionField(Rxy)
-    Rx.<x> = PolynomialRing(F)
-    Fx = FractionField(Rx)
-    FxRy.<y> = PolynomialRing(Fx)
-    g = Fxy(g)
-    g1 = g.numerator()
-    g2 = g.denominator()
-    print('aa', FxRy(g2))
-    
-    
-F = GF(3).algebraic_closure()
-R.<x, y> = PolynomialRing(F, 2)
-g = x
-reduction(g)
+p = 5
+#F.<a> = GF(p^2, 'a')
+F.<a> = GF(p^8, 'a')
+#F = GF(p).algebraic_closure()
+#a = F.gen(2)
+Rx.<x> = PolynomialRing(F)
+P1 = superelliptic(x, 1)
+m = 4
+s = (p^8 - 1)/(p^2 - 1)
+b = a^s
+C = as_cover(P1, [(P1.x)^(m), b*(P1.x)^(m)], prec=300)
+print(C)
+A, B = group_action_matrices_dR(C)
+print('matrices')
+print(magma_module_decomposition(A, B, matrices=False))

tests.sage

@@ -1,16 +1,16 @@
 #load('init.sage')
-print("Expansion at infty test:")
-load('superelliptic/tests/expansion_at_infty.sage')
-print("superelliptic form coordinates test:")
-load('superelliptic/tests/form_coordinates_test.sage')
-print("p-th root test:")
-load('superelliptic/tests/pth_root_test.sage')
+#print("Expansion at infty test:")
+#load('superelliptic/tests/expansion_at_infty.sage')
+#print("superelliptic form coordinates test:")
+#load('superelliptic/tests/form_coordinates_test.sage')
+#print("p-th root test:")
+#load('superelliptic/tests/pth_root_test.sage')
 #print("not working! superelliptic p rank test:")
 #load('superelliptic/tests/p_rank_test.sage')
-print("a-number test:")
-load('superelliptic/tests/a_number_test.sage')
-#print("as_cover_test:")
-#load('as_covers/tests/as_cover_test.sage')
+#print("a-number test:")
+#load('superelliptic/tests/a_number_test.sage')
+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')
 print("dual_element_test:")