Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017). Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, Mukhopadhyay (2022)]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

翻译：有理恒等性检验（RIT）是判定非交换有理公式是否在自由斜域中计算零元的决策问题。该问题在白盒环境下存在确定性多项式时间算法[Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)]，并在黑盒环境下通过自由斜域上线性矩阵的奇异性检验存在随机多项式时间算法[Derksen and Makam (2017)]。事实上，白盒RIT的随机NC算法可由Derksen与Makam (2017)的结果直接推导。设计RIT的高效确定性黑盒算法并理解其并行复杂度是当前领域的主要公开难题。尽管自Garg、Gurvits、Oliveira与Wigderson (2016)的工作以来该问题一直悬而未决，相关研究进展极为有限。实际上，目前唯一已知结果是为仅含二次逆高度的有理公式构造拟多项式规模命中集[Arvind, Chatterjee, Mukhopadhyay (2022)]。本文显著改进了该问题的黑盒复杂度，首次为所有多项式规模有理公式构造出拟多项式规模命中集。我们的构造还导出了白盒RIT的首个确定性拟NC上界。