From d8cfd4af4c6800e7744ab3a6cf9231dd725c9549 Mon Sep 17 00:00:00 2001 From: jgarnek Date: Wed, 26 Oct 2022 15:43:09 +0000 Subject: [PATCH] dzialajacy rozklad na nierozkladalne magma w AS; dwa sposoby na wspolrzedne - jeden sredni --- ...ntary_covers_of_superelliptic_curves.ipynb | 1514 +++++++++-------- 1 file changed, 778 insertions(+), 736 deletions(-) diff --git a/elementary_covers_of_superelliptic_curves.ipynb b/elementary_covers_of_superelliptic_curves.ipynb index 25b42ca..8ff7310 100644 --- a/elementary_covers_of_superelliptic_curves.ipynb +++ b/elementary_covers_of_superelliptic_curves.ipynb @@ -725,6 +725,8 @@ " self.quotient = C\n", " self.functions = list_of_fcts\n", " self.height = len(list_of_fcts)\n", + " F = C.base_ring\n", + " self.base_ring = F\n", " p = C.characteristic\n", " self.characteristic = p\n", " self.prec = prec\n", @@ -733,8 +735,8 @@ " r = f.degree()\n", " delta = GCD(m, r)\n", " self.nb_of_pts_at_infty = delta\n", - " Rxy. = PolynomialRing(GF(p), 2)\n", - " Rt. = LaurentSeriesRing(GF(p), default_prec=prec)\n", + " Rxy. = PolynomialRing(F, 2)\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", "\n", " all_x_series = []\n", " all_y_series = []\n", @@ -786,15 +788,24 @@ " gY = self.quotient.genus()\n", " n = self.height\n", " delta = self.nb_of_pts_at_infty\n", - " return p*gY + (p^n - 1)*(delta - 1) + sum(p^(n-j-1)*(jumps[i][j]-1)*(p-1)/2 for j in range(n) for i in range(delta))\n", + " p = self.characteristic\n", + " return p^n*gY + (p^n - 1)*(delta - 1) + sum(p^(n-j-1)*(jumps[i][j]-1)*(p-1)/2 for j in range(n) for i in range(delta))\n", " \n", " def exponent_of_different(self, i = 0):\n", " jumps = self.jumps\n", " n = self.height\n", " delta = self.nb_of_pts_at_infty\n", + " p = self.characteristic\n", " return sum(p^(n-j-1)*(jumps[i][j]+1)*(p-1) for j in range(n))\n", "\n", - " def holomorphic_differentials_basis(self):\n", + " def exponent_of_different_prim(self, i = 0):\n", + " jumps = self.jumps\n", + " n = self.height\n", + " delta = self.nb_of_pts_at_infty\n", + " p = self.characteristic\n", + " return sum(p^(n-j-1)*(jumps[i][j])*(p-1) for j in range(n))\n", + " \n", + " def holomorphic_differentials_basis(self, parameter = 8):\n", " from itertools import product\n", " x_series = self.x\n", " y_series = self.y\n", @@ -805,21 +816,22 @@ " n = self.height\n", " prec = self.prec\n", " C = self.quotient\n", + " F = self.base_ring\n", " m = C.exponent\n", " r = C.polynomial.degree()\n", " variable_names = 'x, y'\n", " for i in range(n):\n", " variable_names += ', z' + str(i)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", " x, y = Rxyz.gens()[:2]\n", " z = Rxyz.gens()[2:]\n", " RxyzQ = FractionField(Rxyz)\n", - " Rt. = LaurentSeriesRing(GF(p), default_prec=prec)\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", " #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y\n", " S = []\n", " RQxyz = FractionField(Rxyz)\n", " pr = [list(GF(p)) for _ in range(n)]\n", - " for i in range(0, 3*r):\n", + " for i in range(0, parameter*r):\n", " for j in range(0, m):\n", " for k in product(*pr):\n", " eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)\n", @@ -835,12 +847,110 @@ " if len(forms) < self.genus():\n", " print(\"I haven't found all forms.\")\n", " return forms\n", + " \n", + " ##find fcts with pole order in infty's at most pole_order\n", + " def at_most_poles(self, pole_order):\n", + " from itertools import product\n", + " x_series = self.x\n", + " y_series = self.y\n", + " z_series = self.z\n", + " delta = self.nb_of_pts_at_infty\n", + " p = self.characteristic\n", + " n = self.height\n", + " prec = self.prec\n", + " C = self.quotient\n", + " F = self.base_ring\n", + " m = C.exponent\n", + " r = C.polynomial.degree()\n", + " variable_names = 'x, y'\n", + " for i in range(n):\n", + " variable_names += ', z' + str(i)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " RxyzQ = FractionField(Rxyz)\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", + " #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y\n", + " S = []\n", + " RQxyz = FractionField(Rxyz)\n", + " pr = [list(GF(p)) for _ in range(n)]\n", + " for i in range(0, 10*r):\n", + " for j in range(0, m):\n", + " for k in product(*pr):\n", + " eta = as_function(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))*y^j)\n", + " eta_exp = eta.expansion_at_infty()\n", + " S += [(eta, eta_exp)]\n", "\n", + " forms = holomorphic_combinations_fcts(S, pole_order)\n", + "\n", + " for i in range(1, delta):\n", + " forms = [(omega, omega.expansion_at_infty(i = i)) for omega in forms]\n", + " forms = holomorphic_combinations_fcts(forms, pole_order)\n", + " \n", + " return forms\n", + " \n", + " def magical_element(self):\n", + " list_of_elts = self.at_most_poles(self.exponent_of_different_prim())\n", + " result = []\n", + " for a in list_of_elts:\n", + " if a.trace().function != 0:\n", + " result += [a]\n", + " return result\n", + "\n", + " def pseudo_magical_element(self):\n", + " list_of_elts = self.at_most_poles(self.exponent_of_different())\n", + " result = []\n", + " for a in list_of_elts:\n", + " if a.trace().function != 0:\n", + " result += [a]\n", + " return result\n", + " \n", + " ##find fcts with pole order in infty's at most pole_order\n", + " def at_most_poles_forms(self, pole_order):\n", + " from itertools import product\n", + " x_series = self.x\n", + " y_series = self.y\n", + " z_series = self.z\n", + " delta = self.nb_of_pts_at_infty\n", + " p = self.characteristic\n", + " n = self.height\n", + " prec = self.prec\n", + " C = self.quotient\n", + " F = self.base_ring\n", + " m = C.exponent\n", + " r = C.polynomial.degree()\n", + " variable_names = 'x, y'\n", + " for i in range(n):\n", + " variable_names += ', z' + str(i)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " RxyzQ = FractionField(Rxyz)\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", + " #Tworzymy zbiór S form z^i x^j y^k dx/y o waluacji >= waluacja z^(p-1)*dx/y\n", + " S = []\n", + " RQxyz = FractionField(Rxyz)\n", + " pr = [list(GF(p)) for _ in range(n)]\n", + " for i in range(0, 12*r):\n", + " for j in range(0, m):\n", + " for k in product(*pr):\n", + " eta = as_form(self, x^i * prod(z[i1]^(k[i1]) for i1 in range(n))/y^j)\n", + " eta_exp = eta.expansion_at_infty()\n", + " S += [(eta, eta_exp)]\n", + "\n", + " forms = holomorphic_combinations_forms(S, pole_order)\n", + "\n", + " for i in range(1, delta):\n", + " forms = [(omega, omega.expansion_at_infty(i = i)) for omega in forms]\n", + " forms = holomorphic_combinations_forms(forms, pole_order)\n", + " \n", + " return forms\n", + " \n", "#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt\n", "def holomorphic_combinations(S):\n", " C_AS = S[0][0].curve\n", " p = C_AS.characteristic\n", - " F = GF(p)\n", + " F = C_AS.base_ring\n", " prec = C_AS.prec\n", " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", " RtQ = FractionField(Rt)\n", @@ -853,7 +963,7 @@ " list_coeffs = a*[0] + eta_exp.list() + (-minimal_valuation)*[0]\n", " list_coeffs = list_coeffs[:-minimal_valuation]\n", " list_of_lists += [list_coeffs]\n", - " M = matrix(GF(p), list_of_lists)\n", + " M = matrix(F, list_of_lists)\n", " V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S\n", "\n", "\n", @@ -873,25 +983,106 @@ "class as_function:\n", " def __init__(self, C, g):\n", " self.curve = C\n", + " F = C.base_ring\n", " n = C.height\n", " variable_names = 'x, y'\n", " for i in range(n):\n", " variable_names += ', z' + str(i)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", " x, y = Rxyz.gens()[:2]\n", " z = Rxyz.gens()[2:]\n", " RxyzQ = FractionField(Rxyz)\n", " self.function = RxyzQ(g)\n", - " \n", + "\n", + " def __repr__(self):\n", + " return str(self.function)\n", + "\n", + " def __add__(self, other):\n", + " C = self.curve\n", + " g1 = self.function\n", + " g2 = other.function\n", + " return as_function(C, g1 + g2)\n", + "\n", + " def __sub__(self, other):\n", + " C = self.curve\n", + " g1 = self.function\n", + " g2 = other.function\n", + " return as_function(C, g1 - g2)\n", + "\n", + " def __rmul__(self, constant):\n", + " C = self.curve\n", + " g = self.function\n", + " return as_function(C, constant*g)\n", + "\n", + " def expansion_at_infty(self, i = 0):\n", + " C = self.curve\n", + " delta = C.nb_of_pts_at_infty\n", + " F = C.base_ring\n", + " x_series = C.x[i]\n", + " y_series = C.y[i]\n", + " z_series = C.z[i]\n", + " n = C.height\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " RxyzQ = FractionField(Rxyz)\n", + " prec = C.prec\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", + " g = self.function\n", + " sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}\n", + " return g.substitute(sub_list)\n", + "\n", + " def group_action(self, ZN_tuple):\n", + " C = self.curve\n", + " n = C.height\n", + " F = C.base_ring\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " RxyzQ = FractionField(Rxyz)\n", + " sub_list = {x : x, y : y} | {z[j] : z[j]+ZN_tuple[j] for j in range(n)}\n", + " g = self.function\n", + " return as_form(C, g.substitute(sub_list))\n", + "\n", + " def trace(self):\n", + " C = self.curve\n", + " C_super = C.quotient\n", + " n = C.height\n", + " F = C.base_ring\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " RxyzQ = FractionField(Rxyz)\n", + " g = self.function\n", + " result = RxyzQ(0)\n", + " g_num = Rxyz(numerator(g))\n", + " g_den = Rxyz(denominator(g))\n", + " z = prod(z[i] for i in range(n))^(p-1)\n", + " for a in g_num.monomials():\n", + " if (z.divides(a)):\n", + " result += g_num.monomial_coefficient(a)*a/z\n", + " result /= g_den\n", + " Rxy. = PolynomialRing(F, 2)\n", + " return superelliptic_function(C_super, Rxy(result))\n", " \n", "class as_form:\n", " def __init__(self, C, g):\n", " self.curve = C\n", " n = C.height\n", + " F = C.base_ring\n", " variable_names = 'x, y'\n", " for i in range(n):\n", " variable_names += ', z' + str(i)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", " x, y = Rxyz.gens()[:2]\n", " z = Rxyz.gens()[2:]\n", " RxyzQ = FractionField(Rxyz)\n", @@ -903,6 +1094,7 @@ " def expansion_at_infty(self, i = 0):\n", " C = self.curve\n", " delta = C.nb_of_pts_at_infty\n", + " F = C.base_ring\n", " x_series = C.x[i]\n", " y_series = C.y[i]\n", " z_series = C.z[i]\n", @@ -911,12 +1103,12 @@ " variable_names = 'x, y'\n", " for j in range(n):\n", " variable_names += ', z' + str(j)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", " x, y = Rxyz.gens()[:2]\n", " z = Rxyz.gens()[2:]\n", " RxyzQ = FractionField(Rxyz)\n", " prec = C.prec\n", - " Rt. = LaurentSeriesRing(GF(p), default_prec=prec)\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", " g = self.form\n", " sub_list = {x : x_series, y : y_series} | {z[j] : z_series[j] for j in range(n)}\n", " return g.substitute(sub_list)*dx_series\n", @@ -941,10 +1133,11 @@ " def group_action(self, ZN_tuple):\n", " C = self.curve\n", " n = C.height\n", + " F = C.base_ring\n", " variable_names = 'x, y'\n", " for j in range(n):\n", " variable_names += ', z' + str(j)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", " x, y = Rxyz.gens()[:2]\n", " z = Rxyz.gens()[2:]\n", " RxyzQ = FractionField(Rxyz)\n", @@ -956,6 +1149,7 @@ " C = self.curve\n", " n = C.height\n", " gC = C.genus()\n", + " F = C.base_ring\n", " holo = C.holomorphic_differentials_basis()\n", " nb_of_coeffs = 2*gC\n", " holo_exp = [omega.expansion_at_infty() for omega in holo]\n", @@ -966,7 +1160,7 @@ " added_list = a*[0] + added_list\n", " added_list = added_list[:nb_of_coeffs]\n", " list_of_coeffs += [added_list]\n", - " M = matrix(GF(p), list_of_coeffs)\n", + " M = matrix(F, list_of_coeffs)\n", " M = M.transpose()\n", " omega = self.expansion_at_infty()\n", " a = omega.valuation()\n", @@ -976,27 +1170,45 @@ " v = vector(v)\n", " v1 = M.solve_right(v)\n", " return v1\n", + " \n", + " def coordinates2(self, holo):\n", + " C = self.curve\n", + " n = C.height\n", + " gC = C.genus()\n", + " F = C.base_ring\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " from sage.rings.polynomial.toy_variety import linear_representation\n", + " return linear_representation(Rxyz(self.form), holo)\n", "\n", - "def val_magical(C_AS):\n", - " n = C_AS.height\n", - " variable_names = 'x, y'\n", - " for i in range(n):\n", - " variable_names += ', z' + str(i)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", - " x, y = Rxyz.gens()[:2]\n", - " z = Rxyz.gens()[2:]\n", - " RxyzQ = FractionField(Rxyz)\n", - " C = C_AS.quotient\n", - " forms = C.holomorphic_differentials_basis()\n", - " z_mag = prod(z[i]^(p-1) for i in range(n))\n", - " forms_list = [z_mag*RxyzQ(omega.form) for omega in forms]\n", - " forms_list = [as_form(C_AS, omega) for omega in forms_list]\n", - " forms_list = [omega.expansion_at_infty() for omega in forms_list]\n", - " forms_list = [omega.valuation() for omega in forms_list]\n", - " return min(forms_list)\n", - "\n", - "\n", - "\n", + " def trace(self):\n", + " C = self.curve\n", + " C_super = C.quotient\n", + " n = C.height\n", + " F = C.base_ring\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " RxyzQ = FractionField(Rxyz)\n", + " g = self.form\n", + " result = RxyzQ(0)\n", + " g_num = Rxyz(numerator(g))\n", + " g_den = Rxyz(denominator(g))\n", + " z = prod(z[i] for i in range(n))^(p-1)\n", + " for a in g_num.monomials():\n", + " if (z.divides(a)):\n", + " result += g_num.monomial_coefficient(a)*a/z\n", + " result /= g_den\n", + " Rxy. = PolynomialRing(F, 2)\n", + " return superelliptic_form(C_super, Rxy(result))\n", + " \n", "# Given power_series, find its reverse (g with g \\circ power_series = id) with given precision\n", "\n", "def new_reverse(power_series, prec = 10):\n", @@ -1032,8 +1244,8 @@ " while(power_series.valuation() % p == 0 and power_series.valuation() < 0):\n", " M = -power_series.valuation()/p\n", " coeff = power_series.list()[0] #wspolczynnik a_(-p) w f_AS\n", - " correction += coeff*t^(-M)\n", - " power_series = power_series - coeff*(t^(-p*M) - t^(-M))\n", + " correction += coeff.nth_root(p)*t^(-M)\n", + " power_series = power_series - (coeff*t^(-p*M) - coeff.nth_root(p)*t^(-M))\n", " jump = max(-(power_series.valuation()), 0)\n", " try:\n", " T = ((power_series)^(-1)).nth_root(jump) #T is defined by power_series = 1/T^m\n", @@ -1049,16 +1261,18 @@ "def are_forms_linearly_dependent(set_of_forms):\n", " from sage.rings.polynomial.toy_variety import is_linearly_dependent\n", " C = set_of_forms[0].curve\n", + " F = C.base_ring\n", " n = C.height\n", " variable_names = 'x, y'\n", " for i in range(n):\n", " variable_names += ', z' + str(i)\n", - " Rxyz = PolynomialRing(GF(p), n+2, variable_names)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", " denominators = prod(denominator(omega.form) for omega in set_of_forms)\n", " return is_linearly_dependent([Rxyz(denominators*omega.form) for omega in set_of_forms])\n", "\n", "def group_action_matrices(C_AS):\n", " holo = C_AS.holomorphic_differentials_basis()\n", + " F = C_AS.base_ring\n", " n = C_AS.height\n", " A = [[] for i in range(n)]\n", " for omega in holo:\n", @@ -1069,9 +1283,142 @@ " v1 = omega1.coordinates()\n", " A[i] += [v1]\n", " for i in range(n):\n", - " A[i] = matrix(GF(p), A[i])\n", + " A[i] = matrix(F, A[i])\n", " A[i] = A[i].transpose()\n", - " return A" + " return A\n", + "\n", + "def group_action_matrices2(C_AS):\n", + " F = C_AS.base_ring\n", + " n = C_AS.height\n", + " holo = C_AS.holomorphic_differentials_basis()\n", + " holo_forms = [omega.form for omega in holo]\n", + " denom = LCM([denominator(omega) for omega in holo_forms])\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " holo_forms = [Rxyz(omega*denom) for omega in holo_forms]\n", + " A = [[] for i in range(n)]\n", + " for omega in holo:\n", + " for i in range(n):\n", + " ei = n*[0]\n", + " ei[i] = 1\n", + " omega1 = omega.group_action(ei)\n", + " omega1 = denom * omega1\n", + " v1 = omega1.coordinates2(holo_forms)\n", + " A[i] += [v1]\n", + " for i in range(n):\n", + " A[i] = matrix(F, A[i])\n", + " A[i] = A[i].transpose()\n", + " return A\n", + "\n", + "def group_action_matrices2_log(C_AS):\n", + " F = C_AS.base_ring\n", + " n = C_AS.height\n", + " holo = C_AS.at_most_poles_forms(1)\n", + " holo_forms = [omega.form for omega in holo]\n", + " denom = LCM([denominator(omega) for omega in holo_forms])\n", + " variable_names = 'x, y'\n", + " for j in range(n):\n", + " variable_names += ', z' + str(j)\n", + " Rxyz = PolynomialRing(F, n+2, variable_names)\n", + " x, y = Rxyz.gens()[:2]\n", + " z = Rxyz.gens()[2:]\n", + " holo_forms = [Rxyz(omega*denom) for omega in holo_forms]\n", + " A = [[] for i in range(n)]\n", + " for omega in holo:\n", + " for i in range(n):\n", + " ei = n*[0]\n", + " ei[i] = 1\n", + " omega1 = omega.group_action(ei)\n", + " omega1 = denom * omega1\n", + " v1 = omega1.coordinates2(holo_forms)\n", + " A[i] += [v1]\n", + " for i in range(n):\n", + " A[i] = matrix(F, A[i])\n", + " A[i] = A[i].transpose()\n", + " return A\n", + "\n", + "\n", + "#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt\n", + "def holomorphic_combinations_fcts(S, pole_order):\n", + " C_AS = S[0][0].curve\n", + " p = C_AS.characteristic\n", + " F = C_AS.base_ring\n", + " prec = C_AS.prec\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", + " RtQ = FractionField(Rt)\n", + " minimal_valuation = min([Rt(g[1]).valuation() for g in S])\n", + " if minimal_valuation >= -pole_order:\n", + " return [s[0] for s in S]\n", + " list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S\n", + " for eta, eta_exp in S:\n", + " a = -minimal_valuation + Rt(eta_exp).valuation()\n", + " list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]\n", + " list_coeffs = list_coeffs[:-minimal_valuation - pole_order]\n", + " list_of_lists += [list_coeffs]\n", + " M = matrix(F, list_of_lists)\n", + " V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S\n", + "\n", + "\n", + " # Sprawdzamy, jakim formom odpowiadają elementy V.\n", + " forms = []\n", + " for vec in V.basis():\n", + " forma_holo = as_function(C_AS, 0)\n", + " forma_holo_power_series = Rt(0)\n", + " for vec_wspolrzedna, elt_S in zip(vec, S):\n", + " eta = elt_S[0]\n", + " #eta_exp = elt_S[1]\n", + " forma_holo += vec_wspolrzedna*eta\n", + " #forma_holo_power_series += vec_wspolrzedna*eta_exp\n", + " forms += [forma_holo]\n", + " return forms\n", + "\n", + "#given a set S of (form, corresponding Laurent series at some pt), find their combinations holomorphic at that pt\n", + "def holomorphic_combinations_forms(S, pole_order):\n", + " C_AS = S[0][0].curve\n", + " p = C_AS.characteristic\n", + " F = C_AS.base_ring\n", + " prec = C_AS.prec\n", + " Rt. = LaurentSeriesRing(F, default_prec=prec)\n", + " RtQ = FractionField(Rt)\n", + " minimal_valuation = min([Rt(g[1]).valuation() for g in S])\n", + " if minimal_valuation >= -pole_order:\n", + " return [s[0] for s in S]\n", + " list_of_lists = [] #to będzie lista złożona z list współczynników część nieholomorficznych rozwinięcia form z S\n", + " for eta, eta_exp in S:\n", + " a = -minimal_valuation + Rt(eta_exp).valuation()\n", + " list_coeffs = a*[0] + Rt(eta_exp).list() + (-minimal_valuation)*[0]\n", + " list_coeffs = list_coeffs[:-minimal_valuation - pole_order]\n", + " list_of_lists += [list_coeffs]\n", + " M = matrix(F, list_of_lists)\n", + " V = M.kernel() #chcemy wyzerować części nieholomorficzne, biorąc kombinacje form z S\n", + "\n", + "\n", + " # Sprawdzamy, jakim formom odpowiadają elementy V.\n", + " forms = []\n", + " for vec in V.basis():\n", + " forma_holo = as_form(C_AS, 0)\n", + " forma_holo_power_series = Rt(0)\n", + " for vec_wspolrzedna, elt_S in zip(vec, S):\n", + " eta = elt_S[0]\n", + " #eta_exp = elt_S[1]\n", + " forma_holo += vec_wspolrzedna*eta\n", + " #forma_holo_power_series += vec_wspolrzedna*eta_exp\n", + " forms += [forma_holo]\n", + " return forms\n", + "\n", + "#print only forms that are log at the branch pts, but not holomorphic\n", + "def only_log_forms(C_AS):\n", + " list1 = AS.at_most_poles_forms(0)\n", + " list2 = AS.at_most_poles_forms(1)\n", + " result = []\n", + " for a in list2:\n", + " if not(are_forms_linearly_dependent(list1 + result + [a])):\n", + " result += [a]\n", + " return result" ] }, { @@ -1215,139 +1562,11 @@ }, { "cell_type": "code", - "execution_count": 47, + "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ - { - "data": { - "text/plain": [ - "33" - ] - }, - "execution_count": 47, - "metadata": { - }, - "output_type": "execute_result" - } - ], - "source": [ - "AS.genus()" - ] - }, - { - "cell_type": "code", - "execution_count": 22, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "3\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 0 2 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 0 6 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 0 7 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 0 8 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 0 10 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 1 0 [1, 12] 88\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 1 6 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 1 7 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 1 8 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 1 10 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 2 2 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 2 6 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 2 7 [1, 34] 198\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 3 2 [1, 34] 198\n" - ] - }, { "ename": "KeyboardInterrupt", "evalue": "", @@ -1355,51 +1574,113 @@ "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mKeyboardInterrupt\u001b[0m Traceback (most recent call last)", - "Input \u001b[0;32mIn [22]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 10\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m b \u001b[38;5;129;01min\u001b[39;00m GF(p):\n\u001b[1;32m 11\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m c \u001b[38;5;129;01min\u001b[39;00m GF(p):\n\u001b[0;32m---> 12\u001b[0m f1 \u001b[38;5;241m=\u001b[39m \u001b[43msuperelliptic_function\u001b[49m\u001b[43m(\u001b[49m\u001b[43mC_super\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43my\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mx\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m2\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 13\u001b[0m f2 \u001b[38;5;241m=\u001b[39m superelliptic_function(C_super, a\u001b[38;5;241m*\u001b[39my\u001b[38;5;241m+\u001b[39mb\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m+\u001b[39mc\u001b[38;5;241m*\u001b[39mx\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mInteger(\u001b[38;5;241m2\u001b[39m))\n\u001b[1;32m 14\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n", - "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36msuperelliptic_function.__init__\u001b[0;34m(self, C, g)\u001b[0m\n\u001b[1;32m 209\u001b[0m \u001b[38;5;28;01mclass\u001b[39;00m \u001b[38;5;21;01msuperelliptic_function\u001b[39;00m:\n\u001b[0;32m--> 210\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m__init__\u001b[39m(\u001b[38;5;28mself\u001b[39m, C, g):\n\u001b[1;32m 211\u001b[0m F \u001b[38;5;241m=\u001b[39m C\u001b[38;5;241m.\u001b[39mbase_ring\n\u001b[1;32m 212\u001b[0m Rxy \u001b[38;5;241m=\u001b[39m PolynomialRing(F, Integer(\u001b[38;5;241m2\u001b[39m), names\u001b[38;5;241m=\u001b[39m(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mx\u001b[39m\u001b[38;5;124m'\u001b[39m, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124my\u001b[39m\u001b[38;5;124m'\u001b[39m,)); (x, y,) \u001b[38;5;241m=\u001b[39m Rxy\u001b[38;5;241m.\u001b[39m_first_ngens(\u001b[38;5;241m2\u001b[39m)\n", + "Input \u001b[0;32mIn [3]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 12\u001b[0m f2 \u001b[38;5;241m=\u001b[39m superelliptic_function(C_super, x\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m*\u001b[39mInteger(\u001b[38;5;241m3\u001b[39m))\n\u001b[1;32m 13\u001b[0m AS \u001b[38;5;241m=\u001b[39m as_cover(C_super, [f1, f2], prec\u001b[38;5;241m=\u001b[39mInteger(\u001b[38;5;241m300\u001b[39m))\n\u001b[0;32m---> 14\u001b[0m \u001b[38;5;28mprint\u001b[39m(\u001b[43mAS\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmagical_element\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m)\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mas_cover.magical_element\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 170\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mmagical_element\u001b[39m(\u001b[38;5;28mself\u001b[39m):\n\u001b[0;32m--> 171\u001b[0m list_of_elts \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mat_most_poles\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mexponent_of_different_prim\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 172\u001b[0m result \u001b[38;5;241m=\u001b[39m []\n\u001b[1;32m 173\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m a \u001b[38;5;129;01min\u001b[39;00m list_of_elts:\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mas_cover.at_most_poles\u001b[0;34m(self, pole_order)\u001b[0m\n\u001b[1;32m 159\u001b[0m eta_exp \u001b[38;5;241m=\u001b[39m eta\u001b[38;5;241m.\u001b[39mexpansion_at_infty()\n\u001b[1;32m 160\u001b[0m S \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m [(eta, eta_exp)]\n\u001b[0;32m--> 162\u001b[0m forms \u001b[38;5;241m=\u001b[39m \u001b[43mholomorphic_combinations_fcts\u001b[49m\u001b[43m(\u001b[49m\u001b[43mS\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mpole_order\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 164\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(Integer(\u001b[38;5;241m1\u001b[39m), delta):\n\u001b[1;32m 165\u001b[0m forms \u001b[38;5;241m=\u001b[39m [(omega, omega\u001b[38;5;241m.\u001b[39mexpansion_at_infty(i \u001b[38;5;241m=\u001b[39m i)) \u001b[38;5;28;01mfor\u001b[39;00m omega \u001b[38;5;129;01min\u001b[39;00m forms]\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mholomorphic_combinations_fcts\u001b[0;34m(S, pole_order)\u001b[0m\n\u001b[1;32m 599\u001b[0m eta \u001b[38;5;241m=\u001b[39m elt_S[Integer(\u001b[38;5;241m0\u001b[39m)]\n\u001b[1;32m 600\u001b[0m \u001b[38;5;66;03m#eta_exp = elt_S[1]\u001b[39;00m\n\u001b[0;32m--> 601\u001b[0m forma_holo \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[43mvec_wspolrzedna\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43meta\u001b[49m\n\u001b[1;32m 602\u001b[0m \u001b[38;5;66;03m#forma_holo_power_series += vec_wspolrzedna*eta_exp\u001b[39;00m\n\u001b[1;32m 603\u001b[0m forms \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m [forma_holo]\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/structure/element.pyx:1528\u001b[0m, in \u001b[0;36msage.structure.element.Element.__mul__\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1526\u001b[0m if not err:\n\u001b[1;32m 1527\u001b[0m return (right)._mul_long(value)\n\u001b[0;32m-> 1528\u001b[0m return coercion_model.bin_op(left, right, mul)\n\u001b[1;32m 1529\u001b[0m except TypeError:\n\u001b[1;32m 1530\u001b[0m return NotImplemented\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/structure/coerce.pyx:1242\u001b[0m, in \u001b[0;36msage.structure.coerce.CoercionModel.bin_op\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1240\u001b[0m mul_method = getattr(y, '__r%s__'%op_name, None)\n\u001b[1;32m 1241\u001b[0m if mul_method is not None:\n\u001b[0;32m-> 1242\u001b[0m res = mul_method(x)\n\u001b[1;32m 1243\u001b[0m if res is not None and res is not NotImplemented:\n\u001b[1;32m 1244\u001b[0m return res\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mas_function.__rmul__\u001b[0;34m(self, constant)\u001b[0m\n\u001b[1;32m 285\u001b[0m C \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mcurve\n\u001b[1;32m 286\u001b[0m g \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mfunction\n\u001b[0;32m--> 287\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mas_function\u001b[49m\u001b[43m(\u001b[49m\u001b[43mC\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mconstant\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mg\u001b[49m\u001b[43m)\u001b[49m\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mas_function.__init__\u001b[0;34m(self, C, g)\u001b[0m\n\u001b[1;32m 261\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(n):\n\u001b[1;32m 262\u001b[0m variable_names \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124m, z\u001b[39m\u001b[38;5;124m'\u001b[39m \u001b[38;5;241m+\u001b[39m \u001b[38;5;28mstr\u001b[39m(i)\n\u001b[0;32m--> 263\u001b[0m Rxyz \u001b[38;5;241m=\u001b[39m \u001b[43mPolynomialRing\u001b[49m\u001b[43m(\u001b[49m\u001b[43mF\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mn\u001b[49m\u001b[38;5;241;43m+\u001b[39;49m\u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m2\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mvariable_names\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 264\u001b[0m x, y \u001b[38;5;241m=\u001b[39m Rxyz\u001b[38;5;241m.\u001b[39mgens()[:Integer(\u001b[38;5;241m2\u001b[39m)]\n\u001b[1;32m 265\u001b[0m z \u001b[38;5;241m=\u001b[39m Rxyz\u001b[38;5;241m.\u001b[39mgens()[Integer(\u001b[38;5;241m2\u001b[39m):]\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/rings/polynomial/polynomial_ring_constructor.py:609\u001b[0m, in \u001b[0;36mPolynomialRing\u001b[0;34m(base_ring, *args, **kwds)\u001b[0m\n\u001b[1;32m 607\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m arg \u001b[38;5;129;01min\u001b[39;00m args:\n\u001b[1;32m 608\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m--> 609\u001b[0m k \u001b[38;5;241m=\u001b[39m \u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[43marg\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 610\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mTypeError\u001b[39;00m:\n\u001b[1;32m 611\u001b[0m \u001b[38;5;66;03m# Interpret arg as names\u001b[39;00m\n\u001b[1;32m 612\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m names \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/rings/integer.pyx:678\u001b[0m, in \u001b[0;36msage.rings.integer.Integer.__init__\u001b[0;34m()\u001b[0m\n\u001b[1;32m 676\u001b[0m if '_' in x:\n\u001b[1;32m 677\u001b[0m x = x.replace('_', '')\n\u001b[0;32m--> 678\u001b[0m mpz_set_str_python(self.value, str_to_bytes(x), base)\n\u001b[1;32m 679\u001b[0m return\n\u001b[1;32m 680\u001b[0m \n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/rings/integer.pyx:7112\u001b[0m, in \u001b[0;36msage.rings.integer.mpz_set_str_python\u001b[0;34m()\u001b[0m\n\u001b[1;32m 7110\u001b[0m assert base >= 2\n\u001b[1;32m 7111\u001b[0m if mpz_set_str(z, x, base) != 0:\n\u001b[0;32m-> 7112\u001b[0m raise TypeError(\"unable to convert %r to an integer\" % char_to_str(s))\n\u001b[1;32m 7113\u001b[0m if sign < 0:\n\u001b[1;32m 7114\u001b[0m mpz_neg(z, z)\n", "File \u001b[0;32msrc/cysignals/signals.pyx:310\u001b[0m, in \u001b[0;36mcysignals.signals.python_check_interrupt\u001b[0;34m()\u001b[0m\n", "\u001b[0;31mKeyboardInterrupt\u001b[0m: " ] } ], "source": [ - "p = 11\n", - "m = 2\n", - "Rx. = PolynomialRing(GF(p))\n", - "f = x^7 + x^2 + 1\n", + "p = 5\n", + "m = 3\n", + "#F = GF(p)\n", + "F = GF(p^2, 'a')\n", + "a = F.gen()\n", + "Rx. = PolynomialRing(F)\n", + "f = x^m + 1\n", "C_super = superelliptic(f, m)\n", - "print(C_super.genus())\n", - "Rxy. = PolynomialRing(GF(p))\n", "\n", - "for a in GF(p):\n", - " for b in GF(p):\n", - " for c in GF(p):\n", - " if (b != GF(p)(0) or a!=GF(p)(0)):\n", - " f1 = superelliptic_function(C_super, y*x^2)\n", - " f2 = superelliptic_function(C_super, a*y+b*x+c*x^2)\n", - " try:\n", - " AS = as_cover(C_super, [f1, f2], prec=500)\n", - " if AS.jumps[1] >0 and AS.jumps[1] < 67:\n", - " print(a, b, c, AS.jumps, AS.genus())\n", - " except:\n", - " pass" + "Rxy. = PolynomialRing(F, 2)\n", + "f1 = superelliptic_function(C_super, x)\n", + "f2 = superelliptic_function(C_super, x^3)\n", + "AS = as_cover(C_super, [f1, f2], prec=300)\n", + "print(AS.magical_element())" ] }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 8, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[1,\n", + " z1,\n", + " z1^2,\n", + " z0,\n", + " z0*z1,\n", + " -x*z0^2 + z0*z1^2,\n", + " z0^2,\n", + " z0^2*z1,\n", + " z0^2*z1^2 + x*z0*z1 + x^2,\n", + " y,\n", + " -y*z0^2 + y*z1,\n", + " y*z0,\n", + " x,\n", + " -x*z0^2 + x*z1,\n", + " x*z0]" + ] + }, + "execution_count": 8, + "metadata": { + }, + "output_type": "execute_result" + } + ], + "source": [ + "AS.at_most_poles(14)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ - "Rt. = LaurentSeriesRing(GF(p))" ] }, { "cell_type": "code", - "execution_count": 3, + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[((z0^2*z1^2 + x*z0*z1 + x^2)/y) * dx]" + ] + }, + "execution_count": 35, + "metadata": { + }, + "output_type": "execute_result" + } + ], + "source": [ + "only_log_forms(AS)" + ] + }, + { + "cell_type": "code", + "execution_count": 162, "metadata": { "collapsed": false }, @@ -1408,36 +1689,235 @@ "name": "stdout", "output_type": "stream", "text": [ - "0\n" + "I haven't found all forms.\n", + "[1]\n" + ] + } + ], + "source": [ + "dpp = AS.exponent_of_different_prim()\n", + "print(at_most_poles(AS, dpp))" + ] + }, + { + "cell_type": "code", + "execution_count": 66, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "((z0*z1^2 - z1^2 + (a + 1)*x)/y) * dx\n" + ] + } + ], + "source": [ + "Rxy. = PolynomialRing(F, 4)\n", + "omega = as_form(AS, z0^2*z1^2/y - (a+1)*z0*x/y)\n", + "print(omega - omega.group_action([1, 0]))" + ] + }, + { + "cell_type": "code", + "execution_count": 78, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "a*t^-10 + 2*t^-8 + a*t^-6 + (a + 1)*t^-2 + O(t^288)" + ] + }, + "execution_count": 78, + "metadata": { + }, + "output_type": "execute_result" + } + ], + "source": [ + "g = as_function(AS, z0^2*z1^2+z0*(a+1)*x)\n", + "g.expansion_at_infty(i = 0)" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "t^-10 + O(1)\n" ] } ], "source": [ "p = 5\n", - "m = 1\n", - "Rx. = PolynomialRing(GF(p))\n", - "f = x\n", + "m = 8\n", + "F = GF(p)\n", + "Rx. = PolynomialRing(F)\n", + "f = x^6 + 1\n", "C_super = superelliptic(f, m)\n", - "print(C_super.genus())" + "\n", + "Rxy. = PolynomialRing(F, 2)\n", + "f1 = superelliptic_function(C_super, x*y^2)\n", + "print(f1.expansion_at_infty())\n", + "AS = as_cover(C_super, [f1], prec=500)" ] }, { "cell_type": "code", - "execution_count": 5, + "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ - "Rxy. = PolynomialRing(GF(p), 2)\n", + "lll = AS.holomorphic_differentials_basis()" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [ + ], + "source": [ + "f = AS.holomorphic_differentials_basis()[0].trace()" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[z0^2]" + ] + }, + "execution_count": 48, + "metadata": { + }, + "output_type": "execute_result" + } + ], + "source": [ + "AS.magical_element()" + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "metadata": { + "collapsed": false, + "scrolled": true + }, + "outputs": [ + { + "data": { + "text/plain": [ + "68" + ] + }, + "execution_count": 51, + "metadata": { + }, + "output_type": "execute_result" + } + ], + "source": [ + "AS1.exponent_of_different_prim()" + ] + }, + { + "cell_type": "code", + "execution_count": 54, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[-2*x^10 + 2*x^9*y + 2*x^8 - x^7*y + z0^4*z1^4, -2*x^7*z0^4*z1^4 + 2*x^6*y*z0^4*z1^4 + 2*x^5*z0^4*z1^4 - x^4*y*z0^4*z1^4 + y*z0^4*z1^4, -2*x^7*z0^4*z1^4 + 2*x^6*y*z0^4*z1^4 + 2*x^5*z0^4*z1^4 - x^4*y*z0^4*z1^4 + x*z0^4*z1^4, x^6*z0^4*z1^4 - x^5*y*z0^4*z1^4 + 2*x^4*z0^4*z1^4 + x^10 - x^9*y + x*y*z0^4*z1^4 - x^8 - 2*x^7*y, x^6*z0^4*z1^4 - x^5*y*z0^4*z1^4 + 2*x^4*z0^4*z1^4 + x^2*z0^4*z1^4, 2*x^7*z0^4*z1^4 - 2*x^6*y*z0^4*z1^4 + x^5*z0^4*z1^4 - 2*x^4*y*z0^4*z1^4 + x^2*y*z0^4*z1^4, x^7*z0^4*z1^4 - x^6*y*z0^4*z1^4 + 2*x^5*z0^4*z1^4 + x^3*z0^4*z1^4, 2*x^6*z0^4*z1^4 - 2*x^5*y*z0^4*z1^4 - 2*x^4*z0^4*z1^4 + x^3*y*z0^4*z1^4 - 2*x^10 + 2*x^9*y + 2*x^8 - x^7*y]\n" + ] + }, + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[-x^9*z0*z1^3 + x^8*y*z0*z1^3 + 2*x^7*z0^3*z1^3 - 2*x^6*y*z0^3*z1^3 + 2*x^9*z0^3 - 2*x^8*y*z0^3 - x^10*z1^2 + x^9*y*z1^2 + 2*x^8*z0^2*z1^2 - 2*x^7*y*z0^2*z1^2 - x^10*z1 + x^9*y*z1 + 2*x^9*z0*z1 - 2*x^8*y*z0*z1 - 2*x^7*z0*z1^3 - 2*x^5*z0^3*z1^3 + x^4*y*z0^3*z1^3 + 2*x^10 - 2*x^9*y - x^7*z0^3 - 2*x^8*z1^2 - 2*x^6*z0^2*z1^2 + x^5*y*z0^2*z1^2 + 2*x^8*z1 + x^7*y*z1 - 2*x^7*z0*z1 + x^6*y*z0*z1 - x^5*z0*z1^3 + x^8 - 2*x^7*y + 2*x^5*z0^3 - x^6*z1^2 + z0^4*z1^4 + 2*x^5*y*z1 + 2*x^6 - x^5*y, -2*x^7*z0^4*z1^4 + 2*x^6*y*z0^4*z1^4 + 2*x^5*z0^4*z1^4 - x^4*y*z0^4*z1^4 + x^10*z0*z1 - x^9*y*z0*z1 - 2*x^8*z0*z1 - x^7*y*z0*z1 + y*z0^4*z1^4 - 2*x^5*y*z0*z1, -2*x^7*z0^4*z1^4 + 2*x^6*y*z0^4*z1^4 + 2*x^5*z0^4*z1^4 - x^4*y*z0^4*z1^4 + x*z0^4*z1^4, x^6*z0^4*z1^4 - x^5*y*z0^4*z1^4 - x^9*z0*z1^3 + x^8*y*z0*z1^3 - x^7*z0^3*z1^3 + x^6*y*z0^3*z1^3 + 2*x^9*z0^3 - 2*x^8*y*z0^3 - x^10*z1^2 + x^9*y*z1^2 - x^8*z0^2*z1^2 + x^7*y*z0^2*z1^2 + 2*x^4*z0^4*z1^4 - x^10*z1 + x^9*y*z1 - x^9*z0*z1 + x^8*y*z0*z1 - 2*x^7*z0*z1^3 + x^5*z0^3*z1^3 + 2*x^4*y*z0^3*z1^3 + 2*x^10 - 2*x^9*y - x^7*z0^3 - 2*x^8*z1^2 + x^6*z0^2*z1^2 + 2*x^5*y*z0^2*z1^2 + x*y*z0^4*z1^4 - x^8*z1 - x^7*y*z1 + x^7*z0*z1 + 2*x^6*y*z0*z1 - x^5*z0*z1^3 - 2*x^8 + x^7*y + 2*x^5*z0^3 - x^6*z1^2 - 2*x^6*z1 - 2*x^5*y*z1 + 2*x^6 - 2*x^5*y, x^6*z0^4*z1^4 - x^5*y*z0^4*z1^4 + 2*x^4*z0^4*z1^4 + x^2*z0^4*z1^4, 2*x^7*z0^4*z1^4 - 2*x^6*y*z0^4*z1^4 + x^5*z0^4*z1^4 - 2*x^4*y*z0^4*z1^4 - 2*x^10*z0*z1 + 2*x^9*y*z0*z1 + x^2*y*z0^4*z1^4 - x^8*z0*z1 + 2*x^7*y*z0*z1 - x^5*y*z0*z1, -x^9*z0^2*z1^4 + x^8*y*z0^2*z1^4 + x^7*z0^4*z1^4 - x^6*y*z0^4*z1^4 - x^9*z0^4*z1 + x^8*y*z0^4*z1 - x^10*z0*z1^3 + x^9*y*z0*z1^3 - x^9*z0^2*z1^3 + x^8*y*z0^2*z1^3 - x^10*z0^3 + x^9*y*z0^3 - 2*x^9*z0^4 + 2*x^8*y*z0^4 - x^11*z1^2 + x^10*y*z1^2 - 2*x^9*z0^2*z1^2 + 2*x^8*y*z0^2*z1^2 - 2*x^7*z0^2*z1^4 + 2*x^5*z0^4*z1^4 - 2*x^11*z1 + 2*x^10*y*z1 + 2*x^10*z0*z1 - 2*x^9*y*z0*z1 + 2*x^9*z0^2*z1 - 2*x^8*y*z0^2*z1 - 2*x^7*z0^4*z1 - 2*x^8*z0*z1^3 - 2*x^7*z0^2*z1^3 - 2*x^11 + 2*x^10*y - 2*x^8*z0^3 + x^7*z0^4 - 2*x^9*z1^2 + x^7*z0^2*z1^2 - x^5*z0^2*z1^4 + x^3*z0^4*z1^4 + x^9*z1 + x^8*z0*z1 - 2*x^7*y*z0*z1 - 2*x^7*z0^2*z1 + x^6*y*z0^2*z1 - x^5*z0^4*z1 - x^6*z0*z1^3 - x^5*z0^2*z1^3 - x^9 + 2*x^8*y - x^6*z0^3 - 2*x^5*z0^4 - x^7*z1^2 - 2*x^5*z0^2*z1^2 - 2*x^7*z1 + x^5*y*z0*z1 - 2*x^5*z0^2*z1 + 2*x^4*y*z0^2*z1 - x^6*y, 2*x^6*z0^4*z1^4 - 2*x^5*y*z0^4*z1^4 + 2*x^9*z0*z1^3 - 2*x^8*y*z0*z1^3 + 2*x^7*z0^3*z1^3 - 2*x^6*y*z0^3*z1^3 + x^9*z0^3 - x^8*y*z0^3 + 2*x^10*z1^2 - 2*x^9*y*z1^2 + 2*x^8*z0^2*z1^2 - 2*x^7*y*z0^2*z1^2 - 2*x^4*z0^4*z1^4 + x^3*y*z0^4*z1^4 + 2*x^10*z1 - 2*x^9*y*z1 + 2*x^9*z0*z1 - 2*x^8*y*z0*z1 - x^7*z0*z1^3 - 2*x^5*z0^3*z1^3 + x^4*y*z0^3*z1^3 + x^10 - x^9*y + 2*x^7*z0^3 - x^8*z1^2 - 2*x^6*z0^2*z1^2 + x^5*y*z0^2*z1^2 + 2*x^8*z1 + 2*x^7*y*z1 - 2*x^7*z0*z1 + x^6*y*z0*z1 + 2*x^5*z0*z1^3 - x^8 - 2*x^7*y + x^5*z0^3 + 2*x^6*z1^2 - x^6*z1 - x^5*y*z1 + x^6 - x^5*y]\n" + ] + } + ], + "source": [ + "p = 5\n", + "m = 2\n", + "F = GF(p)\n", + "#F = GF(p^2, 'a')\n", + "#a = F.gen()\n", + "Rx. = PolynomialRing(F)\n", + "f = x^2 + 1\n", + "C_super = superelliptic(f, m)\n", + "\n", + "Rxy. = PolynomialRing(F, 2)\n", "f1 = superelliptic_function(C_super, x^3)\n", - "AS = as_cover(C_super, [f1], prec=200)" + "f2 = superelliptic_function(C_super, x^2)\n", + "AS1 = as_cover(C_super, [f1, f2], prec=300)\n", + "print(AS1.pseudo_magical_element())\n", + "print(AS1.magical_element())" ] }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 55, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "ename": "ZeroDivisionError", + "evalue": "inverse of Mod(0, 5) does not exist", + "output_type": "error", + "traceback": [ + "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", + "\u001b[0;31mZeroDivisionError\u001b[0m Traceback (most recent call last)", + "Input \u001b[0;32mIn [55]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[38;5;66;03m#A, B = group_action_matrices2_log(AS1)\u001b[39;00m\n\u001b[0;32m----> 2\u001b[0m A, B \u001b[38;5;241m=\u001b[39m \u001b[43mgroup_action_matrices2\u001b[49m\u001b[43m(\u001b[49m\u001b[43mAS1\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 3\u001b[0m magmathis(A, B, Integer(\u001b[38;5;241m3\u001b[39m), deg \u001b[38;5;241m=\u001b[39m Integer(\u001b[38;5;241m1\u001b[39m))\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mgroup_action_matrices2\u001b[0;34m(C_AS)\u001b[0m\n\u001b[1;32m 586\u001b[0m omega1 \u001b[38;5;241m=\u001b[39m omega\u001b[38;5;241m.\u001b[39mgroup_action(ei)\n\u001b[1;32m 587\u001b[0m omega1 \u001b[38;5;241m=\u001b[39m denom \u001b[38;5;241m*\u001b[39m omega1\n\u001b[0;32m--> 588\u001b[0m v1 \u001b[38;5;241m=\u001b[39m \u001b[43momega1\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mcoordinates2\u001b[49m\u001b[43m(\u001b[49m\u001b[43mholo_forms\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 589\u001b[0m A[i] \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m [v1]\n\u001b[1;32m 590\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(n):\n", + "Input \u001b[0;32mIn [2]\u001b[0m, in \u001b[0;36mas_form.coordinates2\u001b[0;34m(self, holo)\u001b[0m\n\u001b[1;32m 462\u001b[0m z \u001b[38;5;241m=\u001b[39m Rxyz\u001b[38;5;241m.\u001b[39mgens()[Integer(\u001b[38;5;241m2\u001b[39m):]\n\u001b[1;32m 463\u001b[0m \u001b[38;5;28;01mfrom\u001b[39;00m \u001b[38;5;21;01msage\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mrings\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mpolynomial\u001b[39;00m\u001b[38;5;21;01m.\u001b[39;00m\u001b[38;5;21;01mtoy_variety\u001b[39;00m \u001b[38;5;28;01mimport\u001b[39;00m linear_representation\n\u001b[0;32m--> 464\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mlinear_representation\u001b[49m\u001b[43m(\u001b[49m\u001b[43mRxyz\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mform\u001b[49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mholo\u001b[49m\u001b[43m)\u001b[49m\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/rings/polynomial/toy_variety.py:220\u001b[0m, in \u001b[0;36mlinear_representation\u001b[0;34m(p, polys)\u001b[0m\n\u001b[1;32m 218\u001b[0m n \u001b[38;5;241m=\u001b[39m M\u001b[38;5;241m.\u001b[39mnrows() \u001b[38;5;241m-\u001b[39m \u001b[38;5;241m1\u001b[39m\n\u001b[1;32m 219\u001b[0m offset \u001b[38;5;241m=\u001b[39m M\u001b[38;5;241m.\u001b[39mncols() \u001b[38;5;241m-\u001b[39m M\u001b[38;5;241m.\u001b[39mnrows()\n\u001b[0;32m--> 220\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m [M[n, offset \u001b[38;5;241m+\u001b[39m each] \u001b[38;5;241m/\u001b[39m (\u001b[38;5;241m-\u001b[39mM[n, j]) \u001b[38;5;28;01mfor\u001b[39;00m each \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(\u001b[38;5;28mlen\u001b[39m(polys))]\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/rings/polynomial/toy_variety.py:220\u001b[0m, in \u001b[0;36m\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m 218\u001b[0m n \u001b[38;5;241m=\u001b[39m M\u001b[38;5;241m.\u001b[39mnrows() \u001b[38;5;241m-\u001b[39m \u001b[38;5;241m1\u001b[39m\n\u001b[1;32m 219\u001b[0m offset \u001b[38;5;241m=\u001b[39m M\u001b[38;5;241m.\u001b[39mncols() \u001b[38;5;241m-\u001b[39m M\u001b[38;5;241m.\u001b[39mnrows()\n\u001b[0;32m--> 220\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m [\u001b[43mM\u001b[49m\u001b[43m[\u001b[49m\u001b[43mn\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43moffset\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m+\u001b[39;49m\u001b[43m \u001b[49m\u001b[43meach\u001b[49m\u001b[43m]\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m/\u001b[39;49m\u001b[43m \u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[43mM\u001b[49m\u001b[43m[\u001b[49m\u001b[43mn\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mj\u001b[49m\u001b[43m]\u001b[49m\u001b[43m)\u001b[49m \u001b[38;5;28;01mfor\u001b[39;00m each \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(\u001b[38;5;28mlen\u001b[39m(polys))]\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/structure/element.pyx:1737\u001b[0m, in \u001b[0;36msage.structure.element.Element.__truediv__\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1735\u001b[0m cdef int cl = classify_elements(left, right)\n\u001b[1;32m 1736\u001b[0m if HAVE_SAME_PARENT(cl):\n\u001b[0;32m-> 1737\u001b[0m return (left)._div_(right)\n\u001b[1;32m 1738\u001b[0m if BOTH_ARE_ELEMENT(cl):\n\u001b[1;32m 1739\u001b[0m return coercion_model.bin_op(left, right, truediv)\n", + "File \u001b[0;32m/ext/sage/9.7/src/sage/rings/finite_rings/integer_mod.pyx:2623\u001b[0m, in \u001b[0;36msage.rings.finite_rings.integer_mod.IntegerMod_int._div_\u001b[0;34m()\u001b[0m\n\u001b[1;32m 2621\u001b[0m right_inverse = self.__modulus.inverses[(right).ivalue]\n\u001b[1;32m 2622\u001b[0m if right_inverse is None:\n\u001b[0;32m-> 2623\u001b[0m raise ZeroDivisionError(f\"inverse of Mod({right}, {self.__modulus.sageInteger}) does not exist\")\n\u001b[1;32m 2624\u001b[0m else:\n\u001b[1;32m 2625\u001b[0m return self._new_c((self.ivalue * (right_inverse).ivalue) % self.__modulus.int32)\n", + "\u001b[0;31mZeroDivisionError\u001b[0m: inverse of Mod(0, 5) does not exist" + ] + } + ], + "source": [ + "#A, B = group_action_matrices2_log(AS1)\n", + "A, B = group_action_matrices2(AS1)\n", + "magmathis(A, B, 3, deg = 1)" + ] + }, + { + "cell_type": "code", + "execution_count": 56, "metadata": { "collapsed": false }, @@ -1445,22 +1925,22 @@ { "data": { "text/plain": [ - "[[3]]" + "68" ] }, - "execution_count": 6, + "execution_count": 56, "metadata": { }, "output_type": "execute_result" } ], "source": [ - "AS.jumps" + "AS1.genus()" ] }, { "cell_type": "code", - "execution_count": 7, + "execution_count": 57, "metadata": { "collapsed": false }, @@ -1468,40 +1948,18 @@ { "data": { "text/plain": [ - "[(1) * dx, (z0) * dx, (z0^2) * dx, (x) * dx]" + "196" ] }, - "execution_count": 7, + "execution_count": 57, "metadata": { }, "output_type": "execute_result" } ], "source": [ - "AS.holomorphic_differentials_basis()" - ] - }, - { - "cell_type": "code", - "execution_count": 243, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "9" - ] - }, - "execution_count": 243, - "metadata": { - }, - "output_type": "execute_result" - } - ], - "source": [ - "len(AS.holomorphic_differentials_basis())" + "lll = AS1.holomorphic_differentials_basis()\n", + "len(lll)" ] }, { @@ -1511,34 +1969,10 @@ "collapsed": false }, "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[[1, 4], [1, 4]]\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[(1) * dx, (-z0^2 + z1) * dx, (z0) * dx, (1/y) * dx, (z1/y) * dx, (z1^2/y) * dx, (z0/y) * dx, (z0*z1/y) * dx, ((-x*z0^2 + z0*z1^2)/y) * dx, (z0^2/y) * dx, (z0^2*z1/y) * dx, (x/y) * dx, ((-x*z0^2 + x*z1)/y) * dx, (x*z0/y) * dx]\n" - ] - } ], "source": [ - "p = 3\n", - "m = 2\n", - "Rx. = PolynomialRing(GF(p))\n", - "f = x^m + 1\n", - "C_super = superelliptic(f, m)\n", - "\n", - "Rxy. = PolynomialRing(GF(p), 2)\n", - "f1 = superelliptic_function(C_super, x)\n", - "f2 = superelliptic_function(C_super, x^2)\n", - "AS = as_cover(C_super, [f1, f2], prec=300)\n", - "print(AS.jumps)\n", - "print(AS.holomorphic_differentials_basis())" + "#A, B = group_action_matrices2_log(AS1)\n", + "A, B = group_action_matrices2(AS1)" ] }, { @@ -1549,24 +1983,25 @@ }, "outputs": [ { - "data": { - "text/plain": [ - "14" - ] - }, - "execution_count": 25, - "metadata": { - }, - "output_type": "execute_result" + "name": "stdout", + "output_type": "stream", + "text": [ + "True\n", + "True\n", + "True\n" + ] } ], "source": [ - "AS.genus()" + "n = A.dimensions()[0]\n", + "print(A*B == B*A)\n", + "print(A^p == identity_matrix(n))\n", + "print(B^p == identity_matrix(n))" ] }, { "cell_type": "code", - "execution_count": 16, + "execution_count": 29, "metadata": { "collapsed": false }, @@ -1574,22 +2009,74 @@ { "data": { "text/plain": [ - "2" + "28" ] }, - "execution_count": 16, + "execution_count": 29, "metadata": { }, "output_type": "execute_result" } ], "source": [ - "AS.genus()" + "n" ] }, { "cell_type": "code", - "execution_count": 249, + "execution_count": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + ], + "source": [ + "def magmathis(A, B, p, deg = 1):\n", + " n = A.dimensions()[0]\n", + " A = str(list(A))\n", + " B = str(list(B))\n", + " A = A.replace(\"(\", \"\")\n", + " A = A.replace(\")\", \"\")\n", + " B = B.replace(\"(\", \"\")\n", + " B = B.replace(\")\", \"\")\n", + " result = \"A := MatrixAlgebra;\"\n", + " result += \"M := RModule(RSpace(GF(\"+str(p)+\"^\" + str(deg)+\"),\" + str(n) + \"), A);\"\n", + " result += \"IndecomposableSummands(M);\"\n", + " print(magma_free(result))" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "[\n", + "RModule of dimension 1 over GF(3^2),\n", + "RModule of dimension 1 over GF(3^2),\n", + "RModule of dimension 3 over GF(3^2),\n", + "RModule of dimension 3 over GF(3^2),\n", + "RModule of dimension 6 over GF(3^2),\n", + "RModule of dimension 6 over GF(3^2),\n", + "RModule of dimension 8 over GF(3^2)\n", + "]\n" + ] + } + ], + "source": [ + "magmathis(A, B, 3, deg = 2)" + ] + }, + { + "cell_type": "code", + "execution_count": 23, "metadata": { "collapsed": false }, @@ -1597,490 +2084,45 @@ { "data": { "text/plain": [ - "[(1/y) * dx,\n", - " (z0/y) * dx,\n", - " ((-x^2 + z0^2)/y) * dx,\n", - " ((2*x^3 + 2*x^2*z0 + z0^3)/y) * dx,\n", - " ((2*x^4 - 2*x^3*z0 - x^2*z0^2 + z0^4)/y) * dx,\n", - " (x/y) * dx,\n", - " ((-x^2 + x*z0)/y) * dx,\n", - " ((x^3 - 2*x^2*z0 + x*z0^2)/y) * dx,\n", - " ((-x^4 - 2*x^3*z0 + 2*x^2*z0^2 + x*z0^3)/y) * dx]" + ">> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,\n", + " ^\n", + "Runtime error in 'RModule': Arguments have incompatible coefficient rings\n", + ">> IndecomposableSummands(M);\n", + " ^\n", + "User error: Identifier 'M' has not been declared or assigned" ] }, - "execution_count": 249, + "execution_count": 23, "metadata": { }, "output_type": "execute_result" } ], "source": [ - "AS.holomorphic_differentials_basis()" + "magma_free(\"A := MatrixAlgebra; M := RModule(RSpace(GF(3),29), A); IndecomposableSummands(M);\")" ] }, { "cell_type": "code", - "execution_count": 11, + "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { - "name": "stdout", - "output_type": "stream", - "text": [ - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 0 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 0 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 1 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 1 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 1 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 2 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "0 2 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "0 2 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 0 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 0 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 0 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 1 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 1 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 1 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 2 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 2 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "1 2 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "2 0 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "2 0 1 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "2 0 2 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - "output_type": "stream", - "text": [ - "no 4 -th root; divide by 2\n", - "2 1 0 [[1, 4], [1, 4]]\n", - "0\n" - ] - }, - { - "name": "stdout", - 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GF(p):\n", - " for c in GF(p):\n", - " if (a, b, c) != (0, 0, 0):\n", - " f = x^2 + 1\n", - " C_super = superelliptic(f, m)\n", - " print(C_super.genus())\n", - "\n", - " Rxy. = PolynomialRing(GF(p), 2)\n", - " f1 = superelliptic_function(C_super, x)\n", - " f2 = superelliptic_function(C_super, )\n", - " try:\n", - " AS = as_cover(C_super, [f1, f2], prec=500)\n", - " except:\n", - " pass\n", - " print(a, b, c, AS.jumps)" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "metadata": { - "collapsed": false - }, - "outputs": [ - ], - "source": [ - ] - }, - { - "cell_type": "code", - "execution_count": 26, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "[1 2 1 0 0 0 0 0 0 0 0 0 0 0]\n", - "[0 1 0 0 0 0 0 0 0 0 0 0 0 0]\n", - "[0 1 1 0 0 0 0 0 0 0 0 0 0 0]\n", - "[0 0 0 1 0 0 1 0 0 1 0 0 0 0]\n", - "[0 0 0 0 1 0 0 1 0 0 1 0 0 0]\n", - "[0 0 0 0 0 1 0 0 1 0 0 0 0 0]\n", - "[0 0 0 0 0 0 1 0 0 2 0 0 0 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\"\\verb\"0|0|1]\"\n", - "\\end{array}\\)" - ], - "text/latex": [ - "$\\displaystyle \\begin{array}{l}\n", - "\\verb\"[1\"\\verb\" \"\\verb\"1\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"1\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb|[-----+-----+---+---+---+-+-]|\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|1\"\\verb\" \"\\verb\"1\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"1\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb|[-----+-----+---+---+---+-+-]|\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|1\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb|[-----+-----+---+---+---+-+-]|\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|1\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"1|0\"\\verb\" \"\\verb\"0|0|0]\"\\\\\n", - "\\verb|[-----+-----+---+---+---+-+-]|\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|1\"\\verb\" \"\\verb\"1|0|0]\"\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"1|0|0]\"\\\\\n", - "\\verb|[-----+-----+---+---+---+-+-]|\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|1|0]\"\\\\\n", - "\\verb|[-----+-----+---+---+---+-+-]|\\\\\n", - "\\verb\"[0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0\"\\verb\" \"\\verb\"0|0|1]\"\n", - "\\end{array}$" - ], - "text/plain": [ - "'[1 1 0|0 0 0|0 0|0 0|0 0|0|0]\\n[0 1 1|0 0 0|0 0|0 0|0 0|0|0]\\n[0 0 1|0 0 0|0 0|0 0|0 0|0|0]\\n[-----+-----+---+---+---+-+-]\\n[0 0 0|1 1 0|0 0|0 0|0 0|0|0]\\n[0 0 0|0 1 1|0 0|0 0|0 0|0|0]\\n[0 0 0|0 0 1|0 0|0 0|0 0|0|0]\\n[-----+-----+---+---+---+-+-]\\n[0 0 0|0 0 0|1 1|0 0|0 0|0|0]\\n[0 0 0|0 0 0|0 1|0 0|0 0|0|0]\\n[-----+-----+---+---+---+-+-]\\n[0 0 0|0 0 0|0 0|1 1|0 0|0|0]\\n[0 0 0|0 0 0|0 0|0 1|0 0|0|0]\\n[-----+-----+---+---+---+-+-]\\n[0 0 0|0 0 0|0 0|0 0|1 1|0|0]\\n[0 0 0|0 0 0|0 0|0 0|0 1|0|0]\\n[-----+-----+---+---+---+-+-]\\n[0 0 0|0 0 0|0 0|0 0|0 0|1|0]\\n[-----+-----+---+---+---+-+-]\\n[0 0 0|0 0 0|0 0|0 0|0 0|0|1]'" - ] - }, - "execution_count": 28, - "metadata": { - }, - "output_type": "execute_result" - } - ], - "source": [ - "pretty_print(B.jordan_form().str())" + "hh = \"A := MatrixAlgebra; \n", + "M := RModule(RSpace(GF(3),29), A); \n", + "IndecomposableSummands(M);\"" ] }, {