Weak nullstellensatz describes maximal ideals in polynomial rings over algebraically closed fields at least when the cardinality number of variables is finite. Lang obtained the same conclusion also when the transcendence degree of the field over its prime field exceeds the number of variables (I don't know if "weak nullstellensatz" officially now includes Lang's extension, but for here let's say it does.)

How explicitly can we describe the maximal spectrum of polynomial rings over algebraically closed fields when weak nullstellensatz fails?

morethan seems unlikely - such a description would surely embody a proof of an axiom, BPIT, provably independent of $ZF$ . But that has not prevented the development of a whole literature concerning the structure of $\beta{\Bbb N}$ (including more independence results). So my question which asks for a description of the maximal spectrum need not fall to the difficulty of describing the individual ideals. – David Feldman Feb 2 '11 at 6:04