DeRhamComputation/as_covers/as_reduction.sage

def as_reduction(AS, fct):
    '''Simplify rational function fct as a function in the function field of AS, so that z[i] appear in powers <p and only in numerator'''
    n = AS.height
    F = AS.base_ring
    RxyzQ, Rxyz, x, y, z = AS.fct_field
    p = F.characteristic()
    ff = AS.functions
    ff = [RxyzQ(F.function) for F in ff]
    fct = RxyzQ(fct)
    fct1 = numerator(fct)
    fct2 = denominator(fct)
    denom = as_function(AS, fct2)
    denom_norm = prod(as_function(AS, fct2).group_action(g) for g in AS.group.elts if g != AS.group.one)
    fct1 = Rxyz(fct1*denom_norm.function)
    fct2 = Rxyz(fct2*denom_norm.function)
    f_super = Rxyz(AS.quotient.polynomial)
    m_super = AS.quotient.exponent
    if fct2 != 1:
        return as_reduction(AS, fct1)/as_reduction(AS, fct2)
    
    result = RxyzQ(0)
    change = 0
    for a in fct1.monomials():
        degrees_zi = [a.degree(z[i]) for i in range(n)]
        d_div = [a.degree(z[i])//p for i in range(n)]
        if d_div != n*[0] or a.degree(y) >= m_super:
            change = 1
        d_rem = [a.degree(z[i])%p for i in range(n)]
        monomial = fct1.monomial_coefficient(a)*x^(a.degree(x))*y^(a.degree(y)%m_super)*f_super^(a.degree(y)//m_super)*prod(z[i]^(d_rem[i]) for i in range(n))*prod((z[i]+AS.rhs[i])^(d_div[i]) for i in range(n))
        result += RxyzQ(monomial)        
        
    if change == 0:
        return RxyzQ(result)
    else:
        return as_reduction(AS, RxyzQ(result))