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：有理恒等性测试（RIT）是判定一个非交换有理公式在自由斜域中是否计算为零的决策问题。该问题在白盒设置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)]，并在黑盒设置下通过自由斜域上线性矩阵的奇异性测试存在随机多项式时间算法 [Derksen and Makam (2017)]。事实上，Derksen and Makam (2017) 的结果也给出了RIT在白盒设置下的随机NC算法。设计高效的确定性黑盒RIT算法并理解RIT的并行复杂度是该领域的主要开放问题。尽管自 Garg, Gurvits, Oliveira, and Wigderson (2016) 的工作以来这些问题一直未解，但进展有限。事实上，该方向唯一已知的结果是仅针对逆高度为2的有理公式构建了拟多项式大小的命中集 [Arvind, Chatterjee, Mukhopadhyay (2022)]。在本文中，我们显著改进了该问题的黑盒复杂度，并首次为所有多项式大小的有理公式构建了拟多项式大小的命中集。我们的构造还给出了白盒设置下RIT的第一个确定性准NC上界。