class as_polyform:
    def __init__(self, form, mult):
        self.form = form
        self.curve = form.curve
        self.mult =  mult
        
    def __add__(self, other):
        return as_polyform(self.form + other.form, self.mult)
    
    def __repr__(self):
        return str(self.form) + ' dx⊗' + str(self.mult)
    
    def expansion_at_infty(self):
        return self.form.expansion_at_infty()*(self.curve.dx.expansion_at_infty())^(self.mult)
    
    def coordinates(self, basis = 0):
        """Find coordinates of the given holomorphic form self in terms of the basis forms in a list holo."""
        AS = self.curve
        if basis == 0:
            basis = AS.holomorphic_differentials_basis()
        RxyzQ, Rxyz, x, y, z = AS.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.
        denom = LCM([denominator(omega.form.function) for omega in basis])
        basis = [denom*omega.form.function for omega in basis]
        self_with_no_denominator = denom*self.form.function
        return linear_representation_polynomials(Rxyz(self_with_no_denominator), [Rxyz(omega) for omega in basis])
    
    
def as_holo_polydifferentials_basis(AS, mult, threshold = 8):
    v = AS.dx.valuation()
    result = AS.at_most_poles(mult*v, threshold=threshold)
    return [as_polyform(omega, mult) for omega in result]

as_cover.holo_polydifferentials_basis = as_holo_polydifferentials_basis

def as_canonical_ideal(AS, threshold=8):
    B0 = AS.holomorphic_differentials_basis(threshold=threshold)
    F = AS.base_ring
    g = AS.genus()
    B1 = [(B0[i], B0[j]) for i in range(g) for j in range(g) if i <= j]
    B2 = AS.holo_polydifferentials_basis(2, threshold = threshold)
    g = AS.genus()
    r = len(B2)
    M = matrix(F, g^2, r)
    for i in range(0, len(B1)):
        (a, b) = B1[i]
        c = as_function(AS, a.form*b.form)
        c = as_reduction(AS, c)
        c = as_function(AS, c)
        c = as_polyform(c, 2)
        #return M, c.coordinates(basis=B2)
        M[i, :] = vector(c.coordinates(basis=B2))
    K = M.kernel().basis()
    result = []
    for v in K:
        coeffs = {b : 0 for b in B1}
        for i in range(r):
            if v[i] != 0:
                coeffs[B1[i]] += v[i]
        result += [as_tensor_product_forms(B1, coeffs)]
    return result

as_cover.canonical_ideal = as_canonical_ideal

class as_tensor_product_forms:
    def __init__(self, pairs_of_forms, coeffs):
        self.pairs = pairs_of_forms
        self.coeffs = coeffs #dictionary
        self.curve = pairs_of_forms[0][0].curve
        
    def coordinates(self, basis = 0):
        AS = self.curve
        g = AS.genus()
        F = AS.base_ring
        if basis == 0:
            basis = AS.holomorphic_differentials_basis()
        result = matrix(F, g, g)
        for (omega1, omega2) in self.pairs:
            c = omega1.coordinates(basis = basis)
            d = omega2.coordinates(basis = basis)
            for i in range(g):
                for j in range(g):
                    result[i, j] += self.coeffs[(omega1, omega2)]*c[i]*d[j]
        return result
    
    def __repr__(self):
        result = ''
        for (omega1, omega2) in self.pairs:
            if self.coeffs[(omega1, omega2)] !=0:
                result += str(self.coeffs[(omega1, omega2)]) + ' * ' + str(omega1) + '⊗' + str(omega2) + ' + '
        return result
    
    def polynomial(self):
        AS = self.curve
        F = AS.base_ring
        g = AS.genus()
        M = self.coordinates()
        Rg = PolynomialRing(F, 'X', g)
        X = Rg.gens()
        return sum(M[i, j] * X[i]*X[j] for i in range(g) for j in range(g))
    
    def group_action(self, elt):
        p = self.base_ring.characteristic()
        n = self.height
        elt1 = [p^n - a for a in elt]
        pairs_of_forms2 = [(a.group_action(elt), b.group_action(elt1)) for (a, b) in pairs_of_forms]
        coeffs2 = {(a.group_action(elt), b.group_action(elt1)) : self.coeffs[(a, b)] for (a, b) in pairs_of_forms}
        return as_tensor_product_forms(pairs_of_forms2, self.coeffs)