Abstract
The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables [Hrubeš and Wigderson, 2015]. This rank computation problem has deterministic polynomialtime whitebox algorithms [Ankit Garg et al., 2016; Ivanyos et al., 2018] and a randomized polynomialtime algorithm in the blackbox setting [Harm Derksen and Visu Makam, 2017]. In this paper, we propose a new approach for efficient derandomization of blackbox RIT. Additionally, we obtain results for matrix rank computation over the free skew field and construct efficient linear pencil representations for a new class of rational expressions. More precisely, we show:
 Under the hardness assumption that the ABP (algebraic branching program) complexity of every polynomial identity for the k×k matrix algebra is 2^Ω(k) [Andrej Bogdanov and Hoeteck Wee, 2005], we obtain a subexponentialtime blackbox RIT algorithm for rational formulas of inversion height almost logarithmic in the size of the formula. This can be seen as the first "hardness implies derandomization" type theorem for rational formulas.
 We show that the noncommutative rank of any matrix over the free skew field whose entries have small linear pencil representations can be computed in deterministic polynomial time. While an efficient rank computation was known for matrices with noncommutative formulas as entries [Ankit Garg et al., 2020], we obtain the first deterministic polynomialtime algorithms for rank computation of matrices whose entries are noncommutative ABPs or rational formulas.
 Motivated by the definition given by Bergman [George M Bergman, 1976], we define a new class of rational functions where a rational function of inversion height at most h is defined as a composition of a noncommutative rskewed circuit (equivalently an ABP) with inverses of rational functions of this class of inversion height at most h1 which are also disjoint. We obtain a polynomialsize linear pencil representation for this class which gives a whitebox deterministic polynomialtime identity testing algorithm for the class.
BibTeX  Entry
@InProceedings{arvind_et_al:LIPIcs.ITCS.2023.6,
author = {Arvind, V. and Chatterjee, Abhranil and Ghosal, Utsab and Mukhopadhyay, Partha and Ramya, C.},
title = {{On Identity Testing and Noncommutative Rank Computation over the Free Skew Field}},
booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
pages = {6:16:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772631},
ISSN = {18688969},
year = {2023},
volume = {251},
editor = {Tauman Kalai, Yael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17509},
URN = {urn:nbn:de:0030drops175093},
doi = {10.4230/LIPIcs.ITCS.2023.6},
annote = {Keywords: Algebraic Complexity, Identity Testing, Noncommutative rank}
}
Keywords: 

Algebraic Complexity, Identity Testing, Noncommutative rank 
Collection: 

14th Innovations in Theoretical Computer Science Conference (ITCS 2023) 
Issue Date: 

2023 
Date of publication: 

01.02.2023 