Polynomial bounds for invariant functions separating orbits

Abstract

In a representation of a linear algebraic group G, polynomial invariant functions almost always fail to separate orbits. Unless G is reductive, the ring of invariant polynomials may not be finitely generated. Also the number and complexity of the generators may grow rapidly with the size of the representation. We instead consider an extension of the polynomial ring by introducing a “quasi-inverse” that computes the inverse of a function where defined. With the addition of the quasi-inverse, we write straight line programs defining functions that separate the orbits of any linear algebraic group G. The number of these programs and their length have polynomial bounds in the parameters of the representation.