Black-box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial-time

TL;DR

This paper presents the first deterministic quasipolynomial-time black-box identity testing algorithm for noncommutative rational formulas of inversion height two, advancing the understanding of noncommutative algebraic complexity.

Contribution

It introduces a novel hitting set construction for rational formulas of inversion height two, combining matrix realization theory and cyclic division algebra properties.

Findings

First black-box deterministic quasipolynomial-time algorithm for inversion height two

Hitting set construction uses matrix coefficient realization and cyclic division algebra

Embedding Forbes-Shpilka hitting set into cyclic division algebra of small index

Abstract

Hrube\v{s} and Wigderson (2015) initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, deterministic polynomial-time algorithms are known for this problem following the works of Garg, Gurvits, Oliveira, and Wigderson (2016) and Ivanyos, Qiao, and Subrahmanyam (2018). A central open problem in this area is to design efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this problem for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including key concepts from matrix coefficient realization theory (Vol\v{c}i\v{c}, 2018) and properties of cyclic division algebra (Lam, 2001). En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas (2013) inside a cyclic division algebra of small index.