Polynomial time algorithms in invariant theory for torus actions

TL;DR

This paper develops polynomial time algorithms for orbit classification problems in invariant theory specifically for torus actions, connecting algebraic complexity with combinatorial algorithms.

Contribution

It introduces the first efficient algorithms for orbit equality, closure intersection, and containment problems for commutative group actions, expanding computational invariant theory.

Findings

Polynomial time algorithms for orbit problems in torus actions

Efficient methods for finding separating invariants

Algorithms for generating rational invariants

Abstract

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.