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.

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.

Equations

$$X^{-1}=\{x_1^{-1},x_2^{-1},\dots ,x_n^{-1}\}.$$

$$f=\sum _w{\alpha }_{w}w,{\alpha }_w\in \mathbb{F},$$

$$f(a_1,\dots ,a_n)=0$$

$$\phi (x_{i_1}^{b_1}{x}_{i_2}^{b_2}\cdots x_{i_d}^{b_d})=\prod _{j=1}^d(\text{mathbbm}{1}_{[b_j=1]}\cdot y_{i_{j}j}+\text{mathbbm}{1}_{[b_j=-1]}\cdot z_{i_{j}j}),$$

$$\left[\begin{array}{cc}0& y_{ij}\\ \frac{1}{z_{ij}}& 0\end{array}\right],$$

$$\left[\begin{array}{cc}0& z_{ij}\\ \frac{1}{y_{ij}}& 0\end{array}\right].$$

$$M_i=\left[\begin{array}{cccc}0& y_{i1}& 0& 0\\ \frac{1}{z_{i1}}& 0& 0& 0\\ 0& 0& 0& y_{i2}\\ 0& 0& \frac{1}{z_{i2}}& 0\end{array}\right],M_i^{-1}=\left[\begin{array}{cccc}0& z_{i1}& 0& 0\\ \frac{1}{y_{i1}}& 0& 0& 0\\ 0& 0& 0& z_{i2}\\ 0& 0& \frac{1}{y_{i2}}& 0\end{array}\right].$$

$$N_i=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& i& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],N_i^{-1}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& -i& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],N_i^{b_1}{N}_j^{b_2}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& b_{1}i+b_{2}j& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$$

$${\left(f(M_1,\dots ,M_n)\right)}_{1,2d}=\widehat{\phi (H_d(f))}.$$

$$[\phi (m)]\widehat{\phi (H_d(f))}=[m]f\cdot \prod _{j=1}^{d-1}(b_j\cdot i_j+b_{j+1}\cdot i_{j+1}).$$

$$N_i^{\prime }=\left[\begin{array}{cc}1& i\\ 0& 1\end{array}\right],N_i=\left[\begin{array}{cccccc}1& 0& 0& \dots & 0& 0\\ 0& N_i^{\prime }& 0& \dots & 0& 0\\ 0& 0& N_i^{\prime }& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & N_i^{\prime }& 0\\ 0& 0& 0& \dots & 0& 1\end{array}\right].$$

$$N_i^{\prime -1}=\left[\begin{array}{cc}1& -i\\ 0& 1\end{array}\right],N_i^{-1}=\left[\begin{array}{cccccc}1& 0& 0& \dots & 0& 0\\ 0& N_i^{\prime -1}& 0& \dots & 0& 0\\ 0& 0& N_i^{\prime -1}& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & N_i^{\prime -1}& 0\\ 0& 0& 0& \dots & 0& 1\end{array}\right].$$

$$M_{i,p}^{\prime }=\left[\begin{array}{cc}0& y_{ip}\\ \frac{1}{z_{ip}}& 0\end{array}\right],M_i=\left[\begin{array}{ccccc}{M}_{i,1}^{\prime }& 0& 0& \dots & 0\\ 0& M_{i,2}^{\prime }& 0& \dots & 0\\ 0& 0& M_{i,3}^{\prime }& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & M_{i,d}^{\prime }\end{array}\right].$$

$$M_{i,p}^{\prime -1}=\left[\begin{array}{cc}0& z_{ip}\\ \frac{1}{y_{ip}}& 0\end{array}\right],M_i^{-1}=\left[\begin{array}{ccccc}{M}_{i,1}^{\prime -1}& 0& 0& \dots & 0\\ 0& M_{i,2}^{\prime -1}& 0& \dots & 0\\ 0& 0& M_{i,3}^{\prime -1}& \dots & 0\\ & & & \ddots & \\ ⋮& ⋮& ⋮& & ⋮\\ 0& 0& 0& \dots & M_{i,d}^{\prime -1}\end{array}\right].$$

$${\left(m(M_1,\dots ,M_n)\right)}_{1,2d}=\prod _{j=1}^{d-1}(b_j\cdot i_j+b_{j+1}\cdot i_{j+1})\prod _{j=1}^d([b_j=1]y_{i_{j}j}+[b_j=-1]z_{i_{j}j}).$$

$$S_j^{+}$$

$$S_j^{-}$$

$$N_i^{\prime }=\left[\begin{array}{cccc}1& i& 0& i\\ 0& 1& 0& 0\\ 0& i& 1& i\\ 0& 0& 0& 1\end{array}\right],N_i=\left[\begin{array}{ccccc}{N}_i^{\prime }& 0& 0& \dots & 0\\ 0& N_i^{\prime }& 0& \dots & 0\\ 0& 0& N_i^{\prime }& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & N_i^{\prime }\end{array}\right],$$

$$N_i^{\prime -1}=\left[\begin{array}{cccc}1& -i& 0& -i\\ 0& 1& 0& 0\\ 0& -i& 1& -i\\ 0& 0& 0& 1\end{array}\right],N_i^{-1}=\left[\begin{array}{ccccc}{N}_i^{\prime -1}& 0& 0& \dots & 0\\ 0& N_i^{\prime -1}& 0& \dots & 0\\ 0& 0& N_i^{\prime -1}& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \dots & N_i^{\prime -1}\end{array}\right].$$

$$N_i^{\prime b_1}{N}_j^{\prime b_2}=\left[\begin{array}{cccc}1& (b_{1}i+b_{2}j)& 0& (b_{1}i+b_{2}j)\\ 0& 1& 0& 0\\ 0& (b_{1}i+b_{2}j)& 1& (b_{1}i+b_{2}j)\\ 0& 0& 0& 1\end{array}\right].$$

$$M_{i,j}^{\prime }=\left[\begin{array}{cc}0& y_{ij}\\ \frac{1}{z_{ij}}& 0\end{array}\right],M_{{\xi }_i}^{\prime }=\left[\begin{array}{cc}0& {\xi }_i\\ \frac{1}{{\xi }_i}& 0\end{array}\right],$$

$$M_i=\left[\begin{array}{ccccccc}1& 0& 0& 0& \dots & 0& 0\\ 0& M_{{\xi }_1}& 0& 0& \dots & 0& 0\\ 0& 0& M_{i,1}^{\prime }& 0& \dots & 0& 0\\ 0& 0& 0& M_{{\xi }_2}& \dots & 0& 0\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& 0& \dots & M_{{\xi }_{k^{\prime }+1}}& 0\\ 0& 0& 0& 0& \dots & 0& 1\end{array}\right].$$

$$\hat{m}=\prod _{j=1}^{k^{\prime }+1}{\xi }_j^{{\ell }_j}\cdot \prod _{j=1}^{k^{\prime }}([b_{i_j}=1]y_{i_{j}j}+[b_{i_j}=-1]z_{i_{j}j}).$$

$$[\hat{m}]\hat{f}\ne 0\text{ iff }[m]f\ne 0.$$

$$\sum _{i=1}^{N_1}{c}_i{m}_i+\sum _{j=1}^{N_2}{r}_j=\frac{1}{L}\cdot \left(\sum _{i=1}^{N_1}{c}_i{m}_{i}L+\sum _{j=1}^{N_2}{p}_j\right).$$

$$\text{For each }1\le i<j\le n,b_i{\alpha }_i+b_j{\alpha }_j\ne 0.$$

$$N_i^{\prime }=\left[\begin{array}{cc}1& {\alpha }_i\\ 0& 1\end{array}\right],$$

$$N_i^{\prime }=\left[\begin{array}{cccc}1& {\alpha }_i& 0& {\alpha }_i\\ 0& 1& 0& 0\\ 0& {\alpha }_i& 1& {\alpha }_i\\ 0& 0& 0& 1\end{array}\right].$$

$$g(x_1,x_2,\dots ,x_n)=\prod _{1\le i<j\le n}(x_i+x_j)\cdot (x_i-x_j).$$

Full text

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 $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ 111We use $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ to denote $\mathbb{F}{\langle}x_{1},\ldots,x_{n},x^{-1}_{1},\ldots,x^{-1}_{n}{\rangle}$. where $X=\{x_{1},x_{2},\ldots,x_{n}\}$, a subclass of rational expressions of inversion height one. Our main results are the following.

Given a degree $d$ expression $f$ in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ as a black-box, we obtain a randomized $\operatorname{\mbox{\small\rm poly}}(n,d)$ 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. 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 $\operatorname{\mbox{\small\rm poly}}(n,\log s,\log D)$ time.

1 Introduction

Noncommutative computation is an important sub-area of arithmetic circuit complexity. In the usual arithmetic circuit model for noncommutative computation, the arithmetic operations are addition and multiplication. However, the multiplication gates respect the input order since the variables are noncommuting. Analogous to commutative arithmetic computation, the central questions are to show lower bounds for explicit polynomials and derandomization of polynomial identity testing (PIT) for noncommutative polynomial rings. Exploiting the limited cancellations, strong lower bounds and PIT results are known for noncommutative computations (in contrast to the commutative setting). Nisan[Nis91] has shown that any algebraic branching program (ABP) computing the $n\times n$ noncommutative Determinant or Permanent polynomial requires exponential (in $n$) size. On the PIT front, Raz and Shpilka [RS05] have shown a deterministic polynomial-time PIT for noncommutative ABPs in the white-box model. A quasi-polynomial time derandomization is also known for the black-box model [FS12]. However, for general circuits there are no better results (either lower bound or PIT) than known in the commutative setting.

The randomized polynomial-time PIT algorithm for noncommutative circuits computing a polynomial of polynomially bounded degree [BW05] follows from Amitsur-Levitzki theorem [AL50]. The Amitsur-Levitzki theorem states that a nonzero noncommutative polynomial $p\in\mathbb{F}{\langle}X{\rangle}$ of degree $<2k$ cannot be an identity for the matrix ring $\mathbb{M}_{k}(\mathbb{F})$. Additionally, it is shown that a nonzero noncommutative polynomial does not vanish on matrices of dimension logarithmic in the sparsity of the polynomial, yielding a randomized polynomial time algorithm for noncommutative circuits computing a nonzero polynomial of exponential degree and exponential sparsity [AJMR17].

Hrubeš and Wigderson [HW14] initiated the study of noncommutative computation with inverses. In the commutative world, it suffices to consider additions and multiplications. By Strassen’s result [Str73] (extended to finite fields [HY11]), divisions can be efficiently replaced by polynomially many additions and multiplications. However, divisions in noncommutative computation are more complex [HW14]. In the same paper [HW14] the authors introduce rational identity testing: Given a noncommutative formula involving addition, multiplication and division gates, efficiently check if the resulting rational expression is identically zero in the free skew-field of noncommutative rational functions. They show that the rational identity testing problem reduces to the following SINGULAR problem:

Given a matrix $A_{n\times n}$ where the entries are linear forms over noncommuting variables $\{x_{1},x_{2},\ldots,x_{n}\}$, is $A$ invertible in the free skew-field?

In the white-box model the problem is in deterministic polynomial time, and in randomized polynomial time in the black-box model  [GGOW16, IQS18, DM18]. Specifically, for rational formulas of size $s$, random matrix substitutions of dimension linear in $s$ suffices to test if the rational expression is identically zero [DM18].

The complexity of identity testing for general rational expressions remains open. For example, given a noncommutative circuit involving addition, multiplication and division gates, no efficient algorithm is known to check if the resulting rational expression is identically zero in the free skew-field of noncommutative rational functions. In order to precisely formulate the problem, we define classes of rational expressions based on Bergman’s definition [Ber76] of inversion height which we now recall and elaborate upon with some notation.

Definition 1**.**

[Ber76]* Let $X$ be a set of free noncommuting variables. Polynomials in the free ring $\mathbb{F}{\langle}X{\rangle}$ are defined to be rational expressions of height [math]. A rational expression of height $i+1$ is inductively defined to be a polynomial in rational expressions of height at most $i$, and inverses of such expressions.*

Let $\mathcal{E}_{d,0}$ denote all polynomials of degree at most $d$ in the free ring $\mathbb{F}{\langle}X{\rangle}$. We inductively define rational expressions in $\mathcal{E}_{d,i+1}$ as follows: Let $f_{1},f_{2},\ldots,f_{r}$ and $g_{1},g_{2},\ldots,g_{s}$ be rational expressions in $\mathcal{E}_{d,i}$ in the variables $x_{1},x_{2},\ldots,x_{n}$. Let $f(y_{1},y_{2},\ldots,y_{s},z_{1},z_{2},\ldots,z_{r})$ be a degree-$d$ polynomial in $\mathbb{F}{\langle}X{\rangle}$. Then $f(g_{1},g_{2},\ldots,g_{s},f^{-1}_{1},f^{-1}_{2},\ldots,f^{-1}_{r})$ is a rational expression (of inversion height $i+1$) in $\mathcal{E}_{d,i+1}$.

Black-box identity testing for rational expressions is not well understood in general. Bergman has shown [Ber76, Proposition 5.1] that there are rational expressions that are nonzero over a dense subset of $2\times 2$ matrices but evaluate to zero on dense subsets of $3\times 3$ matrices. This makes it difficult to formulate an Amitsur-Levitzki type of theorem[AL50] for rational expressions.

Remark 1**.**

In this connection, we note that Hrubeš and Wigderson [HW14] have observed that testing if a ‘correct’ rational expression $\Phi$ is not identically zero is equivalent to testing if the rational expression $\Phi^{-1}$ is ‘correct’. I.e. testing if a correct rational expression of inversion height $i$ is identically zero or not can be reduced to testing if a rational expression of inversion height $i+1$ is correct or not. Furthermore, testing if a rational expression of inversion height one is correct can be done by applying (to each inversion operation in this expression) a theorem of Amitsur (see [Row80, LZ09]) which implies that a nonzero degree $2d-1$ noncommutative polynomial evaluated on $d\times d$ matrices will be invertible with high probability. However, this does not yield an efficient randomized identity testing algorithm for rational expressions of inversion height one. Because that seems to require testing correctness of expressions of inversion height two which is a question left open in their paper [HW14, Section 9].

The Free Group Algebra

This motivates the study of black-box identity testing for rational expressions in the free group algebra $\mathbb{F}{\langle}X,X^{-1}{\rangle}$.

We consider expressions in the free group algebra $\mathbb{F}{\langle}X,X^{-1}{\rangle}$, where $(X,X^{-1})^{*}$ denotes the free group generated by the $n$ generators $X=\{x_{1},x_{2},\ldots,x_{n}\}$ and their inverses

[TABLE]

Elements of the free group $(X,X^{-1})^{*}$ are words in $X,X^{-1}$. The only relations satisfied by the generators is $x_{i}x_{i}^{-1}=x_{i}^{-1}x_{i}=1$ for all $i$. Thus, the elements in the free group $(X,X^{-1})^{*}$ are the reduced words which are words to which the above relations are not applicable.

The elements of the free group algebra $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ are $\mathbb{F}$-linear combinations of the form

[TABLE]

where each $w\in(X,X^{-1})^{*}$ is a reduced word. The degree of the expression $f$ is defined as the maximum length of a word $w$ such that $\alpha_{w}\neq 0$. The expression $f$ is said to have sparsity $s$ if there are $s$ many reduced words $w$ such that $\alpha_{w}\neq 0$ in $f$. We also use the notation $[w]f$ to denote the coefficient $\alpha_{w}$ of the reduced word $w$ in the expression $f$.

The free noncommutative ring $\mathbb{F}{\langle}X{\rangle}$ is a subalgebra of $\mathbb{F}{\langle}X,X^{-1}{\rangle}$. Clearly, the elements of $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ are a special case of rational expressions of inversion height one. I.e., we note that:

Proposition 1**.**

$\mathbb{F}{\langle}X,X^{-1}{\rangle}\subset\cup_{d>0}\mathcal{E}_{d,1}$.

Note that the rational expressions in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ allows inverses only of the variables $x_{i}$, whereas the free skew field $\mathbb{F}{\lparenless}X{\rparengtr}$ contains all possible rational expressions (with inverses at any nested level).

Our results

The main goal of the current paper is to obtain black-box identity tests for rational expressions in the free group algebra $\mathbb{F}{\langle}X,X^{-1}{\rangle}$.

Our first result is a generalization of the Amitsur-Levitzki theorem[AL50] to $\mathbb{F}{\langle}X,X^{-1}{\rangle}$. Let $A$ be an associative algebra with identity over $\mathbb{F}$. An expression $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ is an identity for $A$ if

[TABLE]

for all $a_{i}\in A$ such that $a^{-1}_{i}$ is defined for each $i\in[n]$.

Theorem 1**.**

Let $\mathbb{F}$ be any field of characteristic zero and $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ be a nonzero expression of degree $d$. Then $f$ is not an identity for the matrix algebra $\mathbb{M}_{2d}(\mathbb{F})$.

The following corollary is immediate.

Corollary 1** (Black-box identity testing for circuits in free group algebra).**

There is a black-box randomized $\operatorname{\mbox{\small\rm poly}}(n,d)$ identity test for degree $d$ expressions in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$.

If the black-box contains a sparse expression, we show efficient deterministic algorithms for identity testing and interpolation algorithm.

Theorem 2** (Black-box identity testing and reconstruction for sparse expressions in free group algebra).**

Let $\mathbb{F}$ be any field of characteristic zero and $f$ is an expression in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ of degree $d$ and sparsity $s$ given as black-box. Then we can reconstruct $f$ in deterministic $\operatorname{\mbox{\small\rm poly}}(n,d,s)$ time with matrix-valued queries to the black-box.

Our next result is another generalization of the Amitsur-Levitzki theorem [AL50] extending a result of [AJMR17] to free group algebras. We show that a nonzero expression $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ of degree $D$ and sparsity $s$ does not vanish on $O(\log s)$ dimensional matrices. It yields a randomized polynomial-time identity test if the black-box contains an expression $f$ of exponential degree and exponential sparsity.

Theorem 3**.**

Let $\mathbb{F}$ be any field of characteristic zero. Then, a degree-$D$ expression $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ of sparsity $s$ is not an identity for the matrix algebra $\mathbb{M}_{k}(\mathbb{F})$ for $k=O(\log s)$.

Corollary 2** (Black-box identity testing for expoential sparse expressions with exponential degree in free group algebra).**

Given a degree-$D$ expression $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ of sparsity $s$ as black-box, we can check whether $f$ is identically zero or not in randomized $\operatorname{\mbox{\small\rm poly}}(n,\log D,\log s)$ time.

Remark 2**.**

We state our results for fields of characteristic zero only for simplicity. However, by suitable modifications, we can extend our results for fields of positive characteristic.

Organization

The paper is organized as follows. In Section 2, we prove Theorem 1, Corollary 1, and Theorem 2. In Section 3, we prove Theorem 3 and Corollary 2. Finally, in Section 4, we discuss suitable modifications to extend our results over finite fields.

2 A Generalization of Amitsur-Levitzki Theorem for Free Group Algebra

The main idea in our proof is to efficiently encode expressions in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ as polynomials in a suitable commutative ring preserving the identity. Let $\mathbb{F}[Y,Z]$ denote the commutative ring $\mathbb{F}[y_{ij},z_{ij}]_{i\in[n],j\in[d]}$ for $n,d\in\mathbb{N}$, where $Y=\{y_{ij}\mid i\in[n],j\in[d]\}$ and $Z=\{z_{ij}\mid i\in[n],j\in[d]\}$.

Definition 2**.**

Define a map $\varphi:\mathbb{F}{\langle}X,X^{-1}{\rangle}\to\mathbb{F}[Y,Z]$ to be a map such that $\varphi$ is identity on $\mathbb{F}$, and for each reduced word $w=x^{b_{1}}_{i_{1}}x^{b_{2}}_{i_{2}}\cdots x^{b_{d}}_{i_{d}}$,

[TABLE]

where $\mathbbm{1}_{[b_{j}=b]}=1$ if $b_{j}=b$ and $\mathbbm{1}_{[b_{j}=b]}=0$ otherwise.

By linearity the map $\varphi$ is defined on all expressions in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$. We observe the following properties of $\varphi$.

The map $\varphi$ is injective on the reduced words $(X,X^{-1})^{*}$. I.e., it maps each reduced word $w\in(X,X^{-1})^{*}$ to a unique monomial over the commuting variables $Y\cup Z$. 2. 2.

Consequently, $\varphi$ is identity preserving. I.e., an expression $f$ in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ is identically zero if and only if its image $\varphi(f)$ is the zero polynomial in $\mathbb{F}[Y,Z]$. 3. 3.

$\varphi$ preserves the sparsity of the expression. I.e., $f$ in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$ is $s$-sparse iff $\varphi(f)$ in $\mathbb{F}[Y,Z]$ is $s$-sparse. 4. 4.

Given the image $\varphi(f)\in\mathbb{F}[Y,Z]$ in its sparse description (i.e., as a linear combination of monomials), we can efficiently recover the sparse description of $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$.

Given polynomials $f,f^{\prime}\in\mathbb{F}[Y,Z]$, we say $f$ and $f^{\prime}$ are weakly equivalent, if for each monomial $m$, $[m]f=0$ if and only if $[m]f^{\prime}=0$, where $[m]f$ denotes the coefficient of monomial $m$ in $f$.

Given a black-box expression $f$ in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$, we show how to evaluate it on suitable matrices and obtain a polynomial in $\mathbb{F}[Y,Z]$ that is weakly equivalent to $\varphi(f)$ as a specific entry of the resulting matrix. The matrix substitutions are based on automata constructions. Similar ideas have been used earlier to design PIT algorithms for noncommutative polynomials [AMS10]. However, since we are dealing with rational expressions, some difficulties arise. The matrix substitutions for the variables $x_{1},\ldots,x_{n}$ are obtained as the corresponding transition matrices $M_{i}$ of the automaton. The matrix substitution for $x_{i}^{-1}$ will be $M_{i}^{-1}$. Therefore, we need to ensure that the transition matrices $M_{i}$ are invertible and sufficiently structured to be useful for the identity testing.

We first illustrate our construction for an example degree-$2$ expression $f=x_{1}x^{-1}_{2}+x_{2}x^{-1}_{1}$, where $X=\{x_{1},x_{2}\}$.

The basic “building block” for the transition matrix $M_{i}$ is the $2\times 2$ block matrix

[TABLE]

whose inverse is

[TABLE]

When the $2\times 2$ block is the $j^{th}$ diagonal block in $M_{i}$, the corresponding automaton will go from state $2j-1$ to state $2j$ replacing $x_{i}$ by $y_{ij}$ (or if $x_{i}^{-1}$ occurs, it will replace it by $z_{ij}$).

We will keep the transition matrix $M_{i}$ for $x_{i}$ a block diagonal matrix with such $2\times 2$ invertible blocks as the principal minors along the diagonal. In order to ensure this we introduce two new variables $W=\{w_{1},w_{2}\}$ and substitute $x_{i}$ by the word $w_{i}x_{i}w_{i}$ in the expression. This will ensure that we do not have two consecutive $x_{i}$ in the resulting reduced words. In fact, between two $X$ variables (or their inverses) we will have inserted exactly two $W$ variables (or their inverses). Now, we define $M_{i}$ for the above example as

[TABLE]

The corresponding transitions of the automaton is shown in Figure 1.

We now describe the transition matrices $N_{i}$ for $w_{i}$. The matrix $N_{i}$ is also a $4\times 4$ block diagonal matrix. There are three blocks along the diagonal. The first and third are $1\times 1$ blocks of the identity. The second one is a $2\times 2$ block for $w_{i}$-transitions from state $q_{2}$ to state $q_{3}$. It ensures that for any subword $w^{b_{1}}_{1}w^{b_{2}}_{2}$, $b_{i}\in\{1,-1\}$, in the resulting product matrix $N^{b_{1}}_{1}N^{b_{2}}_{2}$ the $(1,2)^{th}$ entry of the $2\times 2$ block is nonzero. The corresponding transitions of the automaton is depicted in Figure 2.

[TABLE]

Hence, evaluating $f(N_{1}M_{1}N_{1},N_{2}M_{2}N_{2})$ we obtain (a polynomial weakly equivalent to) $\varphi(f)$ at the $(1,4)^{th}$ entry. The complete automaton is depicted in figure 3.

We now explain the general construction. For $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ let $H_{\ell}(f)$ denote the degree-$\ell$ homogeneous part of $f$. We will denote by $\widehat{\varphi(H_{\ell}(f))}$ an arbitrary polynomial in $\mathbb{F}[Y,Z]$ weakly equivalent to $\varphi(H_{\ell}(f))$.

Lemma 1**.**

Let $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ be a nonzero expression of degree $d$. There is an $n$-tuple of $2d\times 2d$ matrices $(M_{1},M_{2},\ldots,M_{n})$ whose entries are either scalars, or variables $u\in Y\cup Z$, or their inverses $1/u$, such that

[TABLE]

Furthermore, for each degree-$d$ reduced word of $m=x^{b_{1}}_{i_{1}}x^{b_{2}}_{i_{2}}\cdots x^{b_{d}}_{i_{d}}$ in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$,

[TABLE]

Proof.

Let $e_{ij}$, for $i,j\in[k]$, be the $(i,j)^{th}$ elementary matrix in $\mathbb{M}_{k}(\mathbb{F})$: its $(i,j)^{th}$ entry is $1$ and other entries are [math].

We now define the transition matrices of the NFA for variables $\{w_{i}:1\leq i\leq n\}$ and $\{x_{i}:1\leq i\leq n\}$. For each $i\in[n]$, define $2\times 2$ matrix $N^{\prime}_{i}=e_{11}+e_{22}+i\cdot e_{12}$. Now $N_{i}$ is a $2d\times 2d$ matrix defined as the block diagonal matrix,

[TABLE]

[TABLE]

Each $M_{i},1\leq i\leq n$ is the $2d\times 2d$ block diagonal matrix where each $2\times 2$ block $M^{\prime}_{ij},1\leq j\leq d$ is a $2\times 2$ matrix defined as $M^{\prime}_{i,j}=y_{ij}\cdot e_{12}+\frac{1}{z_{ij}}\cdot e_{21}$. Their inverses have a similar structure.

[TABLE]

[TABLE]

The corresponding NFA is depicted in Figure 4. We substitute each $x_{i_{j}}$ by the $2d\times 2d$ matrix $N_{i_{j}}M_{i_{j}}N_{i_{j}}$. Each $x^{-1}_{i_{j}}$ is substituted by its inverse matrix $N^{-1}_{i_{j}}M^{-1}_{i_{j}}N^{-1}_{i_{j}}$.

Correctness.

Consider a degree-$d$ reduced word $m=x^{b_{1}}_{i_{1}}x^{b_{2}}_{i_{2}}\cdots x^{b_{d}}_{i_{d}}$.

Following the automaton construction of Figure 4, $x^{b_{i}}_{i}$ occurring at position $j$ is substituted by $([\mathbbm{1}_{[b_{i}=1]}y_{ij}+\mathbbm{1}_{[b_{i}=-1]}z_{ij})$. Moreover, for each position $j\in[d-1]$, the adjacent pair $x^{b_{j}}_{i_{j}}x^{b_{j+1}}_{i_{j+1}}$ produces a scalar factor $(b_{j}\cdot i_{j}+b_{j+1}\cdot i_{j+1})$ due to the product $N^{b_{j}}_{i_{j}}N^{b_{j+1}}_{i_{j+1}}$. Consequently, it follows that

[TABLE]

As $\varphi$ is a linear map, the lemma follows. ∎

2.1 Black-box identity testing for circuits in free group algebra

Theorem 1 follows easily from Lemma 1. Lemma 1 says that if $f\in\mathbb{F}{\langle}X,X^{-1}{\rangle}$ is nonzero of degree $d$ then the $(1,2d)$ entry of the matrix $p(N_{1}M_{1}N_{1},\ldots,N_{n}M_{n}N_{n})$ is a nonzero polynomial in $\mathbb{F}[Y,Z]$. Hence $f$ can not be an identity for $M_{2d}(\mathbb{F})$.

It also immediately gives an identity testing algorithm. We can randomly substitute for the variables and apply the Schwartz-Zippel-Demillo-Lipton Theorem [Sch80, Zip79, DL78]. This completes the proof of the Corollary 1.

2.2 Reconstruction of sparse expressions in free group algebra

If the black-box contains an $s$-sparse expression in $\mathbb{F}{\langle}X,X^{-1}{\rangle}$, we give a $\operatorname{\mbox{\small\rm poly}}(s,n,d)$ deterministic interpolation algorithm (which also gives a deterministic identity testing for such expressions). We use a result of Klivans-Spielman [KS01, Theorem11] that constructs a test set in deterministic polynomial time for sparse commutative polynomials, which is used for the interpolation algorithm.

Proof of Theorem 2

Let the black-box expression $f$ be $s$-sparse of degree $d$. By Lemma 1, a polynomial $\widehat{\varphi(H_{d}(p))}$ in $\mathbb{F}[Y,Z]$ is obtained at the $(1,2d)^{th}$ entry of the matrix $f(M_{1},\ldots,M_{n})$, where $M_{i}\in\mathbb{M}_{2d}(\mathbb{F}[Y,Z])$ is as defined in Lemma 1. By Definition 2, $\varphi(f)\in\mathbb{F}[Y,Z]$ is $s$-sparse and has $2nd$ variables. Let $\mathcal{H}_{2nd,d,s}$ be the corresponding test set from [KS01] to interpolate a polynomial of degree $d$ and $s$-sparse over $2nd$ variables. Querying the black-box on $M_{1}(\vec{h}),M_{2}(\vec{h}),\ldots,M_{n}(\vec{h})$ for each $\vec{h}\in\mathcal{H}_{2nd,d,s}$ we can interpolate the commutative polynomial $\widehat{\varphi(H_{d}(f))}$ and obtain an expression for $\widehat{\varphi(H_{d}(f))}=\sum^{s}_{t=1}c_{m_{t}}m_{t}$ as a sum of monomials.

We now need to adjust the extra scalar factors in $\widehat{\varphi(H_{d}(f))}$ to obtain $\varphi(H_{d}(f))$. We can perform this adjustment for each monomial as Lemma 1 shows that the extra scalar factor for the word $m=x^{b_{1}}_{i_{1}}x^{b_{2}}_{i_{2}}\cdots x^{b_{\ell}}_{i_{\ell}}$ is just $\alpha_{m}=\prod_{j=1}^{\ell-1}(b_{j}\cdot i_{j}+b_{j+1}\cdot i_{j+1})$. So the algorithm constructs the expression $\widehat{\varphi(H_{d}(f))}=\sum^{s}_{t=1}\frac{c_{m_{t}}}{\alpha_{m_{t}}}m_{t}$. We can remove the factors $\alpha_{m_{t}}$ for each monomial $m_{t}$ and invert the map $\varphi$ (using the $4^{th}$ property of Definition 2) on every monomial $m_{t}$ to obtain $H_{d}(f)$ as a sum of degree $d$ reduced words. This yields the expression for highest degree homogeneous component of $f$. We can repeat the above procedure on $f-H_{d}(f)$ and reconstruct the remaining homogeneous components of $f$. ∎

3 Black-box Identity Testing for Expressions of Exponential Degree and Exponential Sparsity

In this section, we prove a different generalization of Amitsur-Levitzki theorem [AL50] for free group algebras, based on ideas from [AJMR17]. We show that the dimension of the matrix algebra for which a nonzero input expression $f$ does not vanish is logarithmic in the sparsity of $f$. It yields a randomized $\operatorname{\mbox{\small\rm poly}}(\log D,\log s,n)$ time identity testing algorithm when the black-box contains an expression of degree $D$ and sparsity $s$.

We first recall the notion of isolating index set from [AJMR17].

Definition 3**.**

Let $\mathcal{M}\subseteq\{X,X^{-1}\}^{D}$ be a subset of reduced words of degree $D$. An index set $I\subseteq[D]$ is an isolating index set for $\mathcal{M}$ if there is a word $m\in\mathcal{M}$ such that for each $m^{\prime}\in\mathcal{M}\setminus\{m\}$ there is an index $i\in I$ for which $m[i]\neq m^{\prime}[i]$. I.e. no other word in $\mathcal{M}$ agrees with $m$ on all positions in the index set $I$. We say $m$ is an isolated word.

In the following lemma we show that $\mathcal{M}$ has an isolating index set of size $\log|\mathcal{M}|$. The proof is identical to [AJMR17]. Nevertheless, we give the simple details for completeness because we deal with both variables and their inverses.

Lemma 2**.**

[AJMR17]* Let $\mathcal{M}\subseteq\{X,X^{-1}\}^{D}$ be reduced degree-$D$ words. Then $\mathcal{M}$ has an isolating index set of size $k$ which is bounded by $\log|\mathcal{M}|$.*

Proof.

The words $m\in\mathcal{M}$ are indexed, where $m[i]$ denotes the variable (or the inverse of a variable) in the $i^{th}$ position of $m$. Let $i_{1}\leq D$ be the first index such that not all words agree on the $i_{1}^{th}$ position. Let

[TABLE]

For some $j$, $|S^{+}_{j}|$ or $|S^{-}_{j}|$ is of size at most $|\mathcal{M}|/2$. Let $S^{b}_{i_{1}}$ denote that subset, $b\in\{+,-\}$. We replace $\mathcal{M}$ by $S^{b}_{i_{1}}$ and repeat the same argument for at most $\log|\mathcal{M}|$ steps. Clearly, by this process, we identify a set of indices $I=\{i_{1},\ldots,i_{k}^{\prime}\}$, $k^{\prime}\leq\log|\mathcal{M}|$ such that the set shrinks to a singleton set $\{m\}$. Clearly, $I$ is an isolating index set as witnessed by the isolating word $m$. ∎

Proof of Theorem 3

Let $k=4(k^{\prime}+1)$ where $k^{\prime}$ is the size of the isolating set $I$. As in Section 2, we substitute each $x_{i}$ by $w_{i}x_{i}w_{i}$, where $w_{i},i\in[n]$ are $n$ new variables. The transition matrices for $w_{i}$ and $x_{i}$ are denoted by $N_{i}$ and $M_{i}$ respectively.

For $1\leq i\leq n$, we define $k\times k$ matrix $N_{i}$ as a block diagonal matrix of $k$ many $4\times 4$ matrices $N^{\prime}_{i}$ where $N^{\prime}_{i}=I_{4}+i(e_{12}+e_{34}+e_{32}+e_{14})$.

[TABLE]

[TABLE]

Notice that

[TABLE]

We now define the $k\times k$ transition matrix $M_{i}$ as a block diagonal matrix,

[TABLE]

[TABLE]

These matrices can be seen as the transitions of a suitable NFA. We sketch the construction of this NFA.

Let $I=\{i_{1},\ldots,i_{k^{\prime}}\}$ be an isolating set such that $i_{1}<\ldots<i_{k^{\prime}}$. Intuitively, the NFA does one of two operations on each symbol (a variable or its inverse) of the input expression: a Skip or an Encode. In a Skip stage, the NFA deals with positions that are not part of the (guessed) isolating index set. In this stage, the NFA substitutes the $w_{i}$ variables by suitable scalars (coming from the $N^{\prime}_{i}$ matrices) and $x_{i}$ variables by block variables $\{\xi_{1},\ldots\xi_{k^{\prime}+1}\}$. The NFA nondeterministically decides whether the Skip stage is over and it enters the Encode stage for a guessed index of the isolating set. It substitutes $x_{i}$ and $x^{-1}_{i}$ variables by $y_{ij}$ and $z_{ij}$ respectively. Fig. 5 summarizes the action of the NFA.

Define $\hat{f}$ in $\mathbb{F}(Y,Z,\overline{\xi})$ to be rational function we obtain at the $(1,k)^{th}$222Recall that $k=4(k^{\prime}+1)$ where $k^{\prime}$ is the size of an isolating set. entry by evaluating the expression $f(N_{1}M_{1}N_{1},\ldots,N_{n}M_{n}N_{n})$. Notice that, the isolating word $m$ of degree $D$ will be of following form $m=W_{1}x^{b_{i_{1}}}_{i_{1}}W_{2}x^{b_{i_{2}}}_{i_{2}}\cdots W_{k}^{\prime}x^{b_{i_{k}^{\prime}}}_{i_{k}^{\prime}}W_{k^{\prime}+1}$ where each subword $W_{j}=x^{b_{1}}_{j_{1}}x^{b_{2}}_{j_{2}}\cdots x^{b_{\ell_{j}}}_{j_{\ell_{j}}}$ is of length $\ell_{j}\geq 0$, where some of the $W_{j}$ could be the empty word as well.

We refer to an NFA transition $q_{i}\to q_{j}$ as a forward edge if $i<j$ and a backward edge if $i>j$. We classify the backward edges in three categories based on the substitution on the edge-label. We say, a backward edge is of type A if a variable is substituted by a scalar value; a backward edge is of type B if a variable is substituted by $\frac{1}{\xi_{j}}$ for some $j$; a backward edge is of type C if a variable is substituted by $\frac{1}{y_{ij}}$ or $\frac{1}{z_{ij}}$ for some $i,j$.

Consider a walk of the NFA on an input word $m$ that reaches state $k$ using only type A backward edges. In that case, $m$ is substituted by $\alpha\cdot\hat{m}$ where $\hat{m}$ is a monomial over $\{Y,Z,\xi\}$ of same degree,

[TABLE]

and $\alpha$ is some nonzero constant obtained as a product of $[m]f$ with the scalars obtained as substitutions from the edges involving the $w_{i}$ variables in the Skip stages. Indeed, as we can see from the entries of product matrices $N^{\prime b_{1}}_{i}\cdot N^{\prime b_{2}}_{j}$, where $b_{1},b_{2}\in\{-1,1\}$, the scalar $\alpha$ is a product of $[m]f$ with terms of the form $b_{1}i+b_{2}j$, for $i\neq j$, each of which is nonzero for any reduced word.

Claim 1**.**

[TABLE]

Proof.

It suffices to show that for any word $m^{\prime}\neq m$, where $m^{\prime}$ has degree $\leq D$, no walks of the NFA accepting $m^{\prime}$ generate $\hat{m}$ after substitution. We now argue that no other walks in the NFA can generate $\hat{m}$. For a computation path $J$, the monomial $m_{J}$ in $\hat{f}$ has two parts, let us call it $skip_{J}$ and $encode_{J}$ where $skip_{j}$ is a monomial over $\{\xi_{1},\ldots,\xi_{k^{\prime}+1}\}$ and $encode_{j}$ is a monomial over $\{y_{i,j},z_{i,j}\}_{i\in[n],j\in[k^{\prime}]}$. If the computation path $J$ (which is different from the computation path described above for $\hat{m}$) uses only type A backward edges, then necessarily $m_{J}\neq\hat{m}$ from the definition of isolating index set. This argument is analogous to the argument given in [AJMR17].

Now consider a walk $J$ which involves backward edges of other types. Let us first consider those walks that take backward edges only of type A and type B. Such a walk still produces a monomial over $\{y_{i,j},z_{i,j}\}_{i\in[n],j\in[k^{\prime}]}$ and $\{\xi_{i}\}_{1\leq i\leq k^{\prime}+1}$ because division only by $\xi_{i}$ variables occur in the resulting expression. Since $\hat{m}$ is of highest degree, the total degree of these monomials is strictly lesser than degree of $\hat{m}$. For those walks that take at least one backward edge of type C, a rational expression in $\{y_{i,j},z_{i,j}\}_{i\in[n],j\in[k^{\prime}]}$ and $\{\xi_{i}\}_{1\leq i\leq k^{\prime}+1}$ is produced (as there is division by $y_{ij}$ or $z_{ij}$ variables). As the sum of the degree of the numerator and degree of the numerator is bounded by the total degree, the degree of the numerator is smaller than degree of $\hat{m}$.

Thus the $(1,k)^{th}$ entry of the output matrix is of the form $\sum_{i=1}^{N_{1}}c_{i}m_{i}+\sum_{j=1}^{N_{2}}r_{j}$ where $\{m_{1},\ldots,m_{N_{1}}\}$ are monomials arising from different walks (w.l.o.g. assume that $m_{1}=\hat{m}$) and $\{r_{1},\ldots,r_{N_{2}}\}$ are the rational expressions from the other walks (due to the backward edges of type C). Note that, denominator in each $r_{j}$ is a monomial over $Y,Z$ of degree at most $D$. Let $L=\prod_{i=1}^{n}\prod_{j=1}^{k^{\prime}}y^{D}_{i,j}\cdot z^{D}_{i,j}$. Now, we have,

[TABLE]

Since $\hat{m}L\neq m_{i}L$ for any $i\in\{2,\ldots,N_{1}\}$ and degree of each $p_{j}<$ degree of $\hat{m}L$ for any $j\in\{1,\ldots,N_{2}\}$, the numerator of the final expression is a nonzero polynomial in $\mathbb{F}[Y,Z,\overline{\xi}]$. ∎

The above proof shows that the matrix $f(N_{1}M_{1}N_{1},\ldots,N_{n}M_{n}N_{n})$ is nonzero with rational entries in $\mathbb{F}[Y,Z,\overline{\xi}]$. Each entry is a linear combination of terms of the form $m_{1}/m_{2}$, where $m_{1}$ and $m_{2}$ are monomials in $Y\cup Z\cup\{\xi_{1},\ldots,\xi_{k^{\prime}+1}\}$ of degree bounded by $D$. This completes the proof. ∎

To get an identity testing algorithm, we can do random substitutions.The matrix dimension is $\log s$ and the overall running time of the algorithm is $\operatorname{\mbox{\small\rm poly}}(n,\log s,\log D)$. This also proves Corollary 2. ∎

Remark 3**.**

For algorithmic purposes, we note that Theorem 1 is sometimes preferable to Theorem 3. For instance, the encoding used in Theorem 3 does not preserve the sparsity of the polynomial as required in the sparse reconstruction result (Theorem 2).

4 Adaptation for Fields of Positive Characteristic

Let $\mathbb{F}$ be any finite field of characteristic $p$. We need to ensure that for each word $m$ in the free group algebra, the scalar $\alpha_{m}$ (see Equation 1) produced by the automaton described in Section 2 is not zero in $\mathbb{F}$. Recall that, reading $w^{b_{i}}_{i}w^{b_{j}}_{j}$ for two consecutive positions, the automaton produces a scalar $(b_{i}\cdot i+b_{j}\cdot j)$ where $b_{i},b_{j}\in\{-1,+1\}$. Moreover, this is the only way the automaton produces a scalar and for each $m$, $\alpha_{m}$ is a product of such terms. Hence, all we need to ensure is that for each pair $i,j\in[n]$, $(b_{i}\cdot i+b_{j}\cdot j)\neq 0$. Similarly, it ensures that the scalar produced by the automaton described in Section 3 is non-zero.

We note that, if $p$ is more than $2n$ then each term $(b_{i}\cdot i+b_{j}\cdot j)\neq 0\pmod{p}$ where $b_{i},b_{j}\in\{-1,+1\}$ and $i,j\in[n]$. This results in a dependence on the characteristic of the base field for the analogous statements of Theorems 1, 3 over finite field. Additionally, for Theorem 1, the $(1,2d)^{th}$ entry of the output matrix is a polynomial of degree $d$, and for Theorem 3, the degrees of the numerator polynomials in the rational expression of the output matrix is bounded by some scalar multiple of $nD\log s$. This lower bounds the size of the fields in the application. We summarize the above discussion in the following.

Observation 1**.**

We can obtain results analogous to Theorem 1 and Theorem 3 over finite fields of characteristic more than $2n$ and sizes at least $d+1$ or $\Omega(nD\log s)$ respectively.

However, the algorithms presented in Theorem 2 and Corollaries 1, 2 can be modified to work for finite fields of any characteristic. To this end, we first notice the following simple fact.

Proposition 2**.**

Let $\mathbb{F}$ be a finite field of characteristic $p\leq 2n$. In We can find elements $\alpha_{1},\alpha_{2},\ldots,\alpha_{n}$ from a suitable (deterministically constructed) small extension field $\mathbb{F}^{\prime}$ of $\mathbb{F}$ in deterministic $\operatorname{\mbox{\small\rm poly}}(n)$ time, such that for any $b_{i}\in\{-1,1\},1\leq i\leq n$ we have

[TABLE]

Let $\alpha_{1},\alpha_{2},\ldots,\alpha_{n}\in\mathbb{F}^{\prime}$ as given by the above proposition. We modify the matrix ${N^{\prime}}_{i}$ in the proof of Theorem 2 and Corollary 1 as

[TABLE]

and in Corollary 2 we modify ${N^{\prime}}_{i}$ as

[TABLE]

For each pair $i,j\in[n]$, $(b_{i}\cdot\alpha_{i}+b_{j}\cdot\alpha_{j})\neq 0$ by Proposition 2. Thus, for each word $m$, the scalar $\alpha_{m}$ produced by the automata are nonzero in the extension field $\mathbb{F}^{\prime}$ as well. Furthermore, the test set of [KS01] works for all fields. Hence Theorem 2 holds for all finite fields too. To obtain Corollaries 1 and 2, we need to do the random substitution from suitable small degree extension fields and use Schwartz-Zippel-Demillo-Lipton Theorem [Sch80, Zip79, DL78]. In summary, our algorithms in the paper can be adapted to work over all fields.

Proof of Proposition  2. Define polynomial $g\in\mathbb{F}[x_{1},x_{2},\ldots,x_{n}]$ as

[TABLE]

We substitute $y^{i}$ for $x_{i},1\leq i\leq n$. Then $g(y,y^{2},\ldots,y^{n})=G(y)\in\mathbb{F}[y]$ is a univariate polynomial of degree at most $2n^{3}$. Using standard techniques, in deterministic polynomial time we can construct an extension field $\mathbb{F}^{\prime}$ of $\mathbb{F}$ such that $|\mathbb{F}^{\prime}|$ is of $\operatorname{\mbox{\small\rm poly}}(n)\geq 2n^{3}+1$ size. We can find an element $\alpha\in\mathbb{F}^{\prime}$ such that $G(\alpha)\neq 0$ and set $\alpha_{i}=\alpha^{i},1\leq i\leq n$. ∎