Polynomial Bounds for Invariant Functions Separating Orbits

TL;DR

This paper develops polynomial bounds for constructing functions that can separate orbits in algebraic group representations, using straight line programs with a novel quasi-inverse operation.

Contribution

It introduces a new approach using straight line programs with a quasi-inverse to efficiently construct orbit-separating functions with polynomial complexity.

Findings

Constructible functions can separate orbits with polynomially bounded complexity.

The number and length of straight line programs are polynomially bounded in representation parameters.

The method provides a systematic way to handle non-finitely generated invariant rings.

Abstract

Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow rapidly with the size of the representation. We instead study "constructible" functions defined by straight line programs in the polynomial ring, with a new "quasi-inverse" that computes the inverse of a function where defined. We write straight line programs defining constructible functions that separate the orbits of G. The number of these programs and their length have polynomial bounds in the parameters of the representation.