DeRhamComputation/superelliptic/superelliptic_cech_class.sage

class superelliptic_cech:
    def __init__(self, C, omega, fct):
        self.omega0 = omega
        self.omega8 = omega - fct.diffn()
        self.f = fct
        self.curve = C
        
    def __add__(self, other):
        C = self.curve
        return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)
    
    def __sub__(self, other):
        C = self.curve
        return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)

    def __rmul__(self, constant):
        C = self.curve
        w1 = self.omega0.form
        f1 = self.f.function
        w2 = superelliptic_form(C, constant*w1)
        f2 = superelliptic_function(C, constant*f1)
        return superelliptic_cech(C, w2, f2)    
    
    def __repr__(self):
        return "(" + str(self.omega0) + ", " + str(self.f) + ", " + str(self.omega8) + ")" 
    
    def verschiebung(self):
        C = self.curve
        omega = self.omega0
        F = C.base_ring
        Rx.<x> = PolynomialRing(F)
        return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))
    
    def frobenius(self):
        C = self.curve
        fct = self.f.function
        p = C.characteristic
        Rx.<x> = PolynomialRing(F)
        return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))

    def coordinates(self):
        C = self.curve
        F = C.base_ring
        m = C.exponent
        Rx.<x> = PolynomialRing(F)
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        g = C.genus()
        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):
            result = list(omega.coordinates()) + g*[0]
            result = vector([F(a) for a in result])
            return result
        
        coord = fct.coordinates()
        coord = g*[0] + list(coord)
        coord = vector([F(a) for a in coord])
        aux = self
        for i in range(g, 2*g):
            aux -= coord[i]*basis[i]
        aux_f = decomposition_g0_g8(aux.f)[0]
        aux.omega0 -= aux_f.diffn()
        aux.f = 0*C.x
        aux.omega8 = aux.omega0
        return coord + aux.coordinates()
    
    def is_cocycle(self):
        w0 = self.omega0
        w8 = self.omega8
        fct = self.f
        if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():
            return('w0 & w8')
        if not w0.is_regular_on_U0():
            return('w0')
        if not w8.is_regular_on_Uinfty():
            return('w8')
        if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():
            return 1
        return 0

#Auxilliary. If f = f1/f2 is a rational function, return deg f_1 - deg f_2.
def degree_of_rational_fctn(f, F):
    Rx.<x> = PolynomialRing(F)
    Fx = FractionField(Rx)
    f = Fx(f)
    f1 = f.numerator()
    f2 = f.denominator()
    d1 = f1.degree()
    d2 = f2.degree()
    return(d1 - d2)

#Auxilliary. If f = f1/f2 is a rational function, return (leading coeff of f1)/(leading coeff of f2).
def coeff_of_rational_fctn(f, F):
    Rx.<x> = PolynomialRing(F)
    Fx = FractionField(Rx)
    f = Fx(f)
    if f == Rx(0):
        return 0
    f1 = f.numerator()
    f2 = f.denominator()
    d1 = f1.degree()
    d2 = f2.degree()
    a1 = f1.coefficients(sparse = false)[d1]
    a2 = f2.coefficients(sparse = false)[d2]
    return(a1/a2)

#Auxilliary. Given polynomial f(x) and integer d, return
#coefficient of x^d in f (and 0 is deg(f)<d).
def coff(f, d):
    lista = f.coefficients(sparse = false)
    if len(lista) <= d:
        return 0
    return lista[d]

#Auxilliary. Given polynomial f(x) = \sum_i a_i x^i and integer i, return
#only \sum_{j >= i+1} a_j x^{j - i -1}
def cut(f, i):
    R = f.parent()
    coeff = f.coefficients(sparse = false)
    return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))

def polynomial_part(p, h):
    F = base_ring(parent(h))
    Rx.<x> = PolynomialRing(F)
    h = Rx(h)
    result = Rx(0)
    for i in range(0, h.degree()+1):
        if (i%p) == p-1:
            power = Integer((i-(p-1))/p)
            result += F(h[i]) * x^(power)
    return result

#Find delta-th root of unity in field F
def root_of_unity(F, delta):
    Rx.<x> = PolynomialRing(F)
    cyclotomic = x^(delta) - 1
    for root, a in cyclotomic.roots():
        powers = [root^d for d in delta.divisors() if d!= delta]
        if 1 not in powers:
            return root
        
def preimage(U, V, M): #preimage of subspace U under M
    basis_preimage = M.right_kernel().basis()
    imageU = U.intersection(M.transpose().image())
    basis = imageU.basis()
    for v in basis:
        w = M.solve_right(v)
        basis_preimage = basis_preimage + [w]
    return V.subspace(basis_preimage)

def image(U, V, M):
    basis = U.basis()
    basis_image = []
    for v in basis:
        basis_image += [M*v]
    return V.subspace(basis_image)

def flag(F, V, p, test = 0):
    dim = F.dimensions()[0]
    space = VectorSpace(GF(p), dim)
    flag_subspaces = (dim+1)*[0]
    flag_used = (dim+1)*[0]
    final_type = (dim+1)*['?']
    
    flag_subspaces[dim] = space
    flag_used[dim] = 1
   
    
    while 1 in flag_used:
        index = flag_used.index(1)
        flag_used[index] = 0
        U = flag_subspaces[index]
        U_im = image(U, space, V)
        d_im = U_im.dimension()
        final_type[index] = d_im
        U_pre = preimage(U, space, F)
        d_pre = U_pre.dimension()
        
        if flag_subspaces[d_im] == 0:
            flag_subspaces[d_im] = U_im
            flag_used[d_im] = 1
        
        if flag_subspaces[d_pre] == 0:
            flag_subspaces[d_pre] = U_pre
            flag_used[d_pre] = 1
    
    if test == 1:
        print('(', final_type, ')')
    
    for i in range(0, dim+1):
        if final_type[i] == '?' and final_type[dim - i] != '?':
            i1 = dim - i
            final_type[i] = final_type[i1] - i1 + dim/2
    
    final_type[0] = 0
    
    for i in range(1, dim+1):
        if final_type[i] == '?':
            prev = final_type[i-1]
            if prev != '?' and prev in final_type[i+1:]:
                final_type[i] = prev
                
    for i in range(1, dim+1):
        if final_type[i] == '?':
            final_type[i] = min(final_type[i-1] + 1, dim/2)
    
    if is_final(final_type, dim/2):
        return final_type[1:dim/2 + 1]
    print('error:', final_type[1:dim/2 + 1])
    
def is_final(final_type, dim):
    n = len(final_type)
    if final_type[0] != 0:
        return 0
    
    if final_type[n-1] != dim:
        return 0
    
    for i in range(1, n):
        if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:
            return 0
    return 1