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 talk, I'll present my recent work where 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 a quasi-NC RIT algorithm in the white-box setting.
Joint work with V. Arvind (IMSc) and Partha Mukhopadhyay (CMI).