Rational Identity Testing (RIT) is the decision problem of determining whether or not a given noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg et al., 2016; Ivanyos et. al., 2018; Hamada and Hirai, 2021], and a randomized polynomial-time black-box algorithm [Derksen and Makam, 2017] via singularity testing of linear matrices over the free skew field. Designing a subexponential-time deterministic RIT algorithm in black-box is a major open problem in this area. Despite being open for several years, this question has seen very 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 et al., 2022]. In this paper, we settle this problem and obtain a deterministic quasipolynomial-time RIT algorithm for the general case in the black-box setting. Our algorithm uses ideas from the theory of finite dimensional division algebras, algebraic complexity theory, and the theory of generalized formal power series.

翻译：有理恒等性测试（RIT）是判定给定非交换有理公式是否在自由斜体域中计算为零的决策问题。该问题存在确定性多项式时间白盒算法 [Garg 等人, 2016；Ivanyos 等人, 2018；Hamada 和 Hirai, 2021]，以及通过自由斜体域上线性矩阵的奇异性测试得到的随机多项式时间黑盒算法 [Derksen 和 Makam, 2017]。设计黑盒环境下亚指数时间确定性 RIT 算法是该领域的一个重大开放问题。尽管该问题已开放多年，但进展极为有限。事实上，该方向唯一已知结果是针对仅含高度为二的反转的有理公式构建拟多项式大小的击中集 [Arvind 等人, 2022]。在本文中，我们解决了这一问题，并在黑盒设定下为一般情况获得了一个确定性拟多项式时间 RIT 算法。我们的算法借鉴了有限维除环理论、代数复杂性理论和广义形式幂级数理论的思想。