  Although degree bounds and algorithms for the generators of various invariant
rings have been known for decades, little is known about the cardinality of
minimal generating sets. Estimates of such would provide lower bounds for the
runtime of algorithms that compute invariants. Fix a semisimple linear
algebraic group, choose an irreducible representation of highest weight w, and
consider the irreducible representations of highest weight nw. As n goes to
infinity, the cardinality of a minimal set of generating invariants grows
faster than any polynomial in n. On the other hand, combinatorial methods yield
sub-exponential upper bounds for the growth of generating sets for torus
invariants on the binary forms.
