Landau singularities and higher-order polynomial roots

Landau's work on the singularities of Feynman diagrams suggests that they can only be of three types: either poles, logarithmic divergences, or the roots of quadratic polynomials. On the other hand, many Feynman integrals exist whose singularities involve arbitrarily higher-order polynomial roots. We investigate this apparent paradox using concrete examples involving cube roots and roots of a degree-eight polynomial in four dimensions and roots of a degree-six polynomial in two dimensions and suggest that these higher-order singularities can only be approached via kinematic limits of higher codimension than one, thus evading Landau's argument.