def METHOD_NAME(X):
    """
    Reduce the generator matrix of the module defined by ``X``.
    This is Algorithm 6.4 in [BC2012]_ and relies on the row syzygies of
    the matrix ``X``.
    EXAMPLES::
        sage: from sage.geometry.hyperplane_arrangement.check_freeness import less_generators
        sage: R.<x,y,z> = QQ[]
        sage: m = matrix([[1, 0, 0], [0, z, -1], [0, 0, 0], [0, y, 1]])
        sage: less_generators(m)
        [ 1  0  0]
        [ 0  z -1]
        [ 0  y  1]
    """
    R = X.base_ring()
    syz = fun_fact.ff.syz
    zero = R.zero()
    while True:
        Z = matrix(R, syz(X.transpose()))
        # get_column_independent_unit_positions
        J = range(Z.ncols())
        K = []
        for r in Z.rows():
            for j in J:
                elt = r[j]
                if elt.is_unit():
                    K.append(j)
                    J = [l for l in J if r[l] == zero]
                    break
        if not K: # K is empty
            return X
        Kd = set(range(X.nrows())).difference(K)
        X = X.matrix_from_rows(sorted(Kd))