def decomposition_g0_g8(fct):
    '''Writes fct as a difference g0 - g8, with g0 regular on the affine patch and g8 at the points in infinity.'''
    C = fct.curve
    g = C.genus()
    if fct.coordinates() != g*[0]:
        raise ValueError("The given function cannot be written as g0 - g8.")
    
    Fxy, Rxy, x, y = C.fct_field
    fct = Fxy(fct.function)
    num = fct.numerator()
    den = fct.denominator()
    aux_den = superelliptic_function(C, Rxy(den))
    g0 = superelliptic_function(C, 0)
    g8 = superelliptic_function(C, 0)
    for monomial in num.monomials():
        aux = superelliptic_function(C, monomial)
        if aux.expansion_at_infty().valuation() >= aux_den.expansion_at_infty().valuation():
            g8 += num.monomial_coefficient(monomial)*aux/aux_den
        else:
            g0 += num.monomial_coefficient(monomial)*aux/aux_den
    return (g0, g8)
    
def decomposition_omega0_omega8(omega, prec=50):
    '''Writes omega as a difference omega0 - omega8, with omega0 regular on the affine patch and omega8 at the points in infinity.'''
    C = omega.curve
    Fxy, Rxy, x, y = C.fct_field
    fct = Fxy(omega.form)
    num = fct.numerator()
    den = fct.denominator()
    aux_den = superelliptic_function(C, Rxy(den))
    g0 = superelliptic_function(C, 0)
    g8 = superelliptic_function(C, 0)
    dx_valuation = C.dx.expansion_at_infty(prec=prec).valuation()
    for monomial in num.monomials():
        aux = superelliptic_function(C, monomial)
        if aux.expansion_at_infty(prec=prec).valuation() + dx_valuation >= aux_den.expansion_at_infty(prec=prec).valuation():
            g8 += num.monomial_coefficient(monomial)*aux/aux_den
        else:
            g0 += num.monomial_coefficient(monomial)*aux/aux_den
    g0, g8  = g0*C.dx, g8*C.dx
    if g0.is_regular_on_U0():
        return (g0, g8)
    else:
        raise Error("Something went wrong.")