Georgia Institute of Technology College of Sciences

For the classical actions of the linear algebraic groups in characteristic zero (special linear, orthogonal, symplectic) we calculate the arithmetic rank of Hilbert's nullcone ideals via a general theorem on pure subrings. Then, in positive charactersitic, we discuss topological methods that lead to an inequality between arithmetic rank of the nullcone ideal and the dimension of the invariant ring, that can be used to replace the argument of the pure subring (which is false in positive characteristic). In the outline of his argument we spend most of the time over the complex numbers, for better comfort. This is a report on a paper with J Jeffries, V Pandey and A K Singh.