  Can one tell if an ideal is radical just by looking at the degrees of the
generators? In general, this is hopeless. However, there are special
collections of degrees in multigraded polynomial rings, with the property that
any multigraded ideal generated by elements of those degrees is radical. We
call such a collection of degrees a radical support. In this paper, we give a
combinatorial characterization of radical supports. Our characterization is in
terms of properties of cycles in an associated labelled graph. We also show
that the notion of radical support is closely related to that of
Cartwright-Sturmfels ideals. In fact, any ideal generated by multigraded
generators whose multidegrees form a radical support is a Cartwright-Sturmfels
ideal. Conversely, a collection of degrees such that any multigraded ideal
generated by elements of those degrees is Cartwright-Sturmfels is a radical
support.
