Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing

TL;DR

This paper improves explicit Noether Normalization for matrix invariants under conjugation by weakening the conjecture on polynomial identity testing, leveraging recent advances in hitting sets for read-once oblivious algebraic branching programs, and providing a deterministic parallel algorithm for orbit closure intersection.

Contribution

It weakens Mulmuley's conjecture for PIT needed for explicit Noether Normalization and achieves it unconditionally up to quasipolynomial factors, also providing a deterministic parallel algorithm for orbit closure intersection.

Findings

We improve Mulmuley's reduction and weaken the PIT conjecture.

Quasipolynomial size hitting sets for ROABPs enable explicit normalization.

A deterministic parallel algorithm for orbit closure intersection is developed.

Abstract

Mulmuley recently gave an explicit version of Noether's Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley's reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors, who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena, as PIT algorithms for this model also have implications in Mulmuley's work. Previous work have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich.