# Efficient Black-Box Identity Testing over Free Group Algebra

## TL;DR

This paper develops randomized algorithms for identity testing of noncommutative rational functions in free group algebra, extending classical theorems and providing efficient solutions for a specific subclass of rational expressions.

## Contribution

It introduces randomized and deterministic algorithms for identity testing in free group algebra, generalizing the Amitsur-Levitzki theorem to this setting.

## Key findings

- Randomized polynomial-time algorithm for identity testing in free group algebra.
- Deterministic polynomial-time algorithm based on sparsity for identity testing.
- Extension of Amitsur-Levitzki theorem to noncommutative rational functions.

## Abstract

Hrube\v{s} and Wigderson [HW14] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. It is now known that the problem can be solved in deterministic polynomial time in the white-box model for noncommutative formulas with inverses, and in randomized polynomial time in the black-box model [GGOW16, IQS18, DM18], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions remains open in general (when the formula size is not polynomially bounded). We solve the problem for a natural special case. We consider polynomial expressions in the free group algebra $\mathbb{F}\langle X, X^{-1}\rangle$ where $X=\{x_1, x_2, \ldots, x_n\}$, a subclass of rational expressions of inversion height one. Our main results are the following. 1. Given a degree $d$ expression $f$ in $\mathbb{F}\langle X, X^{-1}\rangle$ as a black-box, we obtain a randomized $\text{poly}(n,d)$ algorithm to check whether $f$ is an identically zero expression or not. We obtain this by generalizing the Amitsur-Levitzki theorem [AL50] to $\mathbb{F}\langle X, X^{-1}\rangle$. This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2. Given an expression $f$ in $\mathbb{F}\langle X, X^{-1}\rangle$ of degree at most $D$, and sparsity $s$, as black-box, we can check whether $f$ is identically zero or not in randomized $\text{poly}(n,\log s, \log D)$ time.