On the Factorization of Non-Commutative Polynomials (in Free Associative   Algebras)

Konrad Schrempf

arXiv:1706.01806·math.RA·August 9, 2018·J. Symb. Comput.

On the Factorization of Non-Commutative Polynomials (in Free Associative Algebras)

Konrad Schrempf

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TL;DR

This paper presents a straightforward method for factorizing non-commutative polynomials in free associative algebras by leveraging their minimal linear representations and solving related polynomial equations.

Contribution

It introduces a novel approach linking factorizations to block structures in system matrices, simplifying the factorization process for non-commutative polynomials.

Findings

01

Establishes a correspondence between factorizations and zero blocks in system matrices.

02

Reduces factorization to solving polynomial equations with commuting unknowns.

03

Provides a practical method for identifying irreducible factors in non-commutative polynomials.

Abstract

We describe a simple approach to factorize non-commutative (nc) polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomial equations (with at most quadratic terms) with commuting unknowns to compute appropriate transformation matrices (if possible).

Equations159

A = z x y . . - x . . 1 = z x 0 001 [1 y 0 - x 01] .

A = z x y . . - x . . 1 = z x 0 001 [1 y 0 - x 01] .

\mu\mathcal{A}_{f}=\bigl{(}u_{f},A_{f},\mu v_{f}\bigr{)}.

\mu\mathcal{A}_{f}=\bigl{(}u_{f},A_{f},\mu v_{f}\bigr{)}.

A_{f} + A_{g} = ([u_{f} .], [A_{f} . - A_{f} u_{f}^{⊤} u_{g} A_{g}], [v_{f} v_{g}]) .

A_{f} + A_{g} = ([u_{f} .], [A_{f} . - A_{f} u_{f}^{⊤} u_{g} A_{g}], [v_{f} v_{g}]) .

A_{f} \cdot A_{g} = ([u_{f} .], [A_{f} . - v_{f} u_{g} A_{g}], [. v_{g}]) .

A_{f} \cdot A_{g} = ([u_{f} .], [A_{f} . - v_{f} u_{g} A_{g}], [. v_{g}]) .

A_{f}^{- 1} = ([1 .], [- v_{f} . A_{f} u_{f}], [. 1]) .

A_{f}^{- 1} = ([1 .], [- v_{f} . A_{f} u_{f}], [. 1]) .

H (p) = . 1 . - 1 1 . - 1 . . - 1 . . - 1 . . . .

H (p) = . 1 . - 1 1 . - 1 . . - 1 . . - 1 . . . .

\hat{A} = [A_{11} A_{21} . A_{22}], \overset{s}{^} = Q^{- 1} s and \overset{v}{^} = P v = v

\hat{A} = [A_{11} A_{21} . A_{22}], \overset{s}{^} = Q^{- 1} s and \overset{v}{^} = P v = v

[I_{n - 1} T^{'} . 1] [A^{'} B] I_{n} = [A^{'} .] .

[I_{n - 1} T^{'} . 1] [A^{'} B] I_{n} = [A^{'} .] .

1 . . - x . 1 . z - 1 s = . . 1 .

1 . . - x . 1 . z - 1 s = . . 1 .

A_{B}^{'} = 1 . . - x . . . z 1

A_{B}^{'} = 1 . . - x . . . z 1

U + (a_{m - 1, j})_{j = m}^{n} - T (a_{ij})_{i, j = m}^{n} = [0 \dots 0] and T (v_{i})_{i = m}^{n} = 0.

U + (a_{m - 1, j})_{j = m}^{n} - T (a_{ij})_{i, j = m}^{n} = [0 \dots 0] and T (v_{i})_{i = m}^{n} = 0.

U + [a_{m - 1, m} \dots a_{m - 1, n}] - T 1 . . . a_{m, m + 1} ⋱ . . \dots ⋱ 1 . a_{m, n} ⋮ a_{n - 1, n} 1 = [0 \dots 0] .

U + [a_{m - 1, m} \dots a_{m - 1, n}] - T 1 . . . a_{m, m + 1} ⋱ . . \dots ⋱ 1 . a_{m, n} ⋮ a_{n - 1, n} 1 = [0 \dots 0] .

[A_{p} . - v_{p} u_{q} A_{q}] s^{'} = [. v_{q}] .

[A_{p} . - v_{p} u_{q} A_{q}] s^{'} = [. v_{q}] .

s^{'} = [A_{p}^{- 1} . A_{p}^{- 1} v_{p} u_{q} A_{q}^{- 1}] [. v_{q}] = [s_{p} q s_{q}] .

s^{'} = [A_{p}^{- 1} . A_{p}^{- 1} v_{p} u_{q} A_{q}^{- 1}] [. v_{q}] = [s_{p} q s_{q}] .

s

s

t

P = I_{m - 2} . . . 1 . . T I_{n - m + 1} and Q = I_{m - 2} . . . 1 . . U I_{n - m + 1},

P = I_{m - 2} . . . 1 . . T I_{n - m + 1} and Q = I_{m - 2} . . . 1 . . U I_{n - m + 1},

A_{p} = (1, [1 . - x 1], 1) and A_{q} = 1, 1 . . y 1 . - 1 - x 1, 1

A_{p} = (1, [1 . - x 1], 1) and A_{q} = 1, 1 . . y 1 . - 1 - x 1, 1

1 . . . . - x 1 . . . . - 1 1 . . . . y 1 . . . - 1 - x 1 s = . . . . 1, s = x (1 - y x) 1 - y x 1 - y x x 1 .

1 . . . . - x 1 . . . . - 1 1 . . . . y 1 . . . - 1 - x 1 s = . . . . 1, s = x (1 - y x) 1 - y x 1 - y x x 1 .

1 . . . . - x 1 . . . - x 01 . . . . y 1 . . . - 1 - x 1 s = . . . . 1, s = x (1 - y x) 0 1 - y x x 1,

1 . . . . - x 1 . . . - x 01 . . . . y 1 . . . - 1 - x 1 s = . . . . 1, s = x (1 - y x) 0 1 - y x x 1,

A = [1 . . .], 1 . . . - x 1 . . 0 y 1 . 0 - 1 - x 1, . . . 1 .

A = [1 . . .], 1 . . . - x 1 . . 0 y 1 . 0 - 1 - x 1, . . . 1 .

A = [1 . . . . .], 1 . . . . . - x 1 . . . . . - y 1 . . . - 1 . . 1 . . . . . - x 1 . . . . - z - y 1, . . - 1 . . 1 .

A = [1 . . . . .], 1 . . . . . - x 1 . . . . . - y 1 . . . - 1 . . 1 . . . . . - x 1 . . . . - z - y 1, . . - 1 . . 1 .

(P, Q) = I_{k - 1} . . . 1 . . T I_{n - k}, I_{k - 1} . . . 1 . . U I_{n - k}

(P, Q) = I_{k - 1} . . . 1 . . T I_{n - k}, I_{k - 1} . . . 1 . . U I_{n - k}

A^{[k]} = [u_{\underline{1}} . .], A_{1, 1} . . A_{1, 2} 1 . A_{1, 3} A_{2, 3} A_{3, 3}, v_{\underline{1}} v_{\underline{2}} v_{\underline{3}}

A^{[k]} = [u_{\underline{1}} . .], A_{1, 1} . . A_{1, 2} 1 . A_{1, 3} A_{2, 3} A_{3, 3}, v_{\underline{1}} v_{\underline{2}} v_{\underline{3}}

P A Q

P A Q

= A_{1, 1} . . A_{1, 2} 1 . A_{1, 2} U + A_{1, 3} U + A_{2, 3} + T A_{3, 3} A_{3, 3},

P v

U + A_{2, 3} + T A_{3, 3} = 0 and v_{\underline{2}} + T v_{\underline{3}} = 0.

U + A_{2, 3} + T A_{3, 3} = 0 and v_{\underline{2}} + T v_{\underline{3}} = 0.

[β_{k + 1} β_{k + 2} β_{k + 3}] + [000] + [α_{k + 1} α_{k + 2} α_{k + 3}] 1 . . - x 1 . - z - y 1

[β_{k + 1} β_{k + 2} β_{k + 3}] + [000] + [α_{k + 1} α_{k + 2} α_{k + 3}] 1 . . - x 1 . - z - y 1

[- 1] + [α_{k + 1} α_{k + 2} α_{k + 3}] . . 1

A_{1} = (u, A, v) = [1 . . . .], 1 . . . . - x 1 . . . - 1 . 1 . . . . - x 1 . . y - z - y 1, . . . . 1 .

A_{1} = (u, A, v) = [1 . . . .], 1 . . . . - x 1 . . . - 1 . 1 . . . . - x 1 . . y - z - y 1, . . . . 1 .

(P, Q) = I_{k - 1} . . T 1 . . . I_{n - k}, I_{k - 1} . . U 1 . . . I_{n - k}

(P, Q) = I_{k - 1} . . T 1 . . . I_{n - k}, I_{k - 1} . . U 1 . . . I_{n - k}

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Full text

On the Factorization of

Non-Commutative Polynomials

(in Free Associative Algebras)

Konrad Schrempf111Contact: [email protected], Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria. Supported by the Austrian FWF Project P25510-N26 “Spectra on Lamplighter groups and Free Probability”

Abstract

We describe a simple approach to factorize non-commutative polynomials, that is, elements in free associative algebras (over a commutative field), into atoms (irreducible elements) based on (a special form of) their minimal linear representations. To be more specific, a correspondence between factorizations of an element and upper right blocks of zeros in the system matrix (of its representation) is established. The problem is then reduced to solving a system of polynomial equations (with at most quadratic terms) with commuting unknowns to compute appropriate transformation matrices (if possible).

Keywords: free associative algebra, factorization of polynomials, minimal linear representation, companion matrix, free field, non-commutative formal power series AMS Classification: Primary 16K40, 16Z05; Secondary 16G99, 16S10

Introduction

Considering non-commutative (nc for short) polynomials (elements of the free associative algebra) as elements in its universal field of fractions (free field) seems —at a first glance— like to take a sledgehammer to crack a nut. It is maybe more common to view them as nc rational series (or nc formal power series). Therefore one could skip the rather complicated theory behind free fields, briefly introduced in [Coh03, Section 9.3]. For details we refer to [Coh95, Section 6.4].

However, as we shall see, the normal form of Cohn and Reutenauer [CR94] provides new insights. For a given element (in the free field) minimal linear representations can be transformed into each other by invertible matrices over the ground field. In our case we are interested in the (finite set of) transformation matrices which yield all possible factorizations (modulo insertion of units and permutation of commuting factors) of a given polynomial.

Here we use the concept of admissible linear systems (ALS for short) of [Coh72], closely related to linear representations. At any step, an ALS (for a nc polynomial) can be easily transformed into a proper linear system [SS78, Section II.1].

A more general concept for the factorization of arbitrary elements in free fields (for example non-commutative rational formal power series) is considered in future work. There (left and right) divisors will be defined on the level of linear representations. In this way additional structure compensates what is missing in fields (when every non-zero element is invertible).

Section 1 provides the notation and the basic (algebraic) setup. Section 2 contains the main result, Theorem 2.24, which describes a correspondence between factorizations and upper right blocks of zeros (in the system matrix). To be able to formulate it, Proposition 2.5 (“minimal” polynomial multiplication) is necessary. The main idea of its proof can be further developed to Algorithm 2.17 (minimizing “polynomial” admissible linear systems). Section 3 generalizes the concept of companion matrices to provide immediately minimal linear representations for a broader class of elements in the free associative algebra. Section 4 illustrates the concept step by step.

To our knowledge the literature on the factorization of elements in free associative algebras is rather sparse. Caruso [Car10] describes ideas of J. Davenport using homogenization. We have not yet investigated how their approach compares to ours. Some special cases (for example variable disjoint factorizations) are treated in [ARJ15].

Here we do not consider factorizations in skew polynomial rings (or rather domains), or —more general— rings satisfying the Ore condition [Coh03, Section 7.1] or [Coh85, Section 0.8]. A starting point in this context is [HL13] or [LH18]. Factorizations of skew polynomials have various connections to other areas, see for example [Ret10] and [GRW01], just to mention two.

1 Representing Non-Commutative Polynomials

There are much simpler ways for representing elements in free associative algebras (in such a way that addition and multiplication can be defined properly) than the following general presented here. However, including the inverse we keep the full flexibility, which could be used later to compute the left (or right) greatest common divisor of two polynomials $p,q$ by minimizing the linear representation for $p^{-1}q$ in which common left factors would cancel.

Notation. The set of the natural numbers is denoted by $\mathbb{N}=\{1,2,\ldots\}$ , that including zero by $\mathbb{N}_{0}$ . Zero entries in matrices are usually replaced by (lower) dots to stress the structure of the non-zero entries unless they result from transformations where there were possibly non-zero entries before. We denote by $I_{n}$ the identity matrix of size $n$ respectively $I$ if the size is clear from the context.

Let $\mathbb{K}$ be a commutative field, $\overline{\mathbb{K}}$ its algebraic closure and $X=\{x_{1},x_{2},\ldots,x_{d}\}$ be a finite alphabet. $\mathbb{K}\langle X\rangle$ denotes the free associative algebra or free $\mathbb{K}$ -algebra (or “algebra of non-commutative polynomials”) and $\mathbb{K}(\!\langle X\rangle\!)$ denotes the universal field of fractions (or “free field”) of $\mathbb{K}\langle X\rangle$ , see [Coh95], [CR99]. In our examples the alphabet is usually $X=\{x,y,z\}$ . Including the algebra of nc rational series $\mathbb{K}^{\text{rat}}\langle\!\langle X\rangle\!\rangle$ we have the following chain of inclusions: $\mathbb{K}\subsetneq\mathbb{K}\langle X\rangle\subsetneq\mathbb{K}^{\text{rat}}\langle\!\langle X\rangle\!\rangle\subsetneq\mathbb{K}(\!\langle X\rangle\!)$ .

The free monoid $X^{*}$ generated by $X$ is the set of all finite words $x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}$ with $i_{k}\in\{1,2,\ldots,d\}$ . An element of the alphabet is called letter, an element of $X^{*}$ is called word. The multiplication on $X^{*}$ is just the concatenation of words, that is, $(x_{i_{1}}\cdots x_{i_{m}})\cdot(x_{j_{1}}\cdots x_{j_{n}})=x_{i_{1}}\cdots x_{i_{m}}x_{j_{1}}\cdots x_{j_{n}}$ , with neutral element $1$ , the empty word. The length of a word $w=x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}$ is $m$ , denoted by $|w|=m$ or $\ell(w)=m$ . For detailled introductions see [BR11, Chapter 1] or [SS78, Section I.1].

Definition 1.1 (Inner Rank, Full Matrix, Hollow Matrix,

see [Coh85], [CR99]).

Given a matrix $A\in\mathbb{K}\langle X\rangle^{n\times n}$ , the inner rank of $A$ is the smallest number $m\in\mathbb{N}$ such that there exists a factorization $A=TU$ with $T\in\mathbb{K}\langle X\rangle^{n\times m}$ and $U\in\mathbb{K}\langle X\rangle^{m\times n}$ . The matrix $A$ is called full if $m=n$ , non-full otherwise. It is called hollow if it contains a zero submatrix of size $k\times l$ with $k+l>n$ .

Definition 1.2 (Associated Matrices,

[Coh95]).

Two matrices $A$ and $B$ over $\mathbb{K}\langle X\rangle$ (of the same size) are called associated over a subring $R\subseteq\mathbb{K}\langle X\rangle$ if there exist (over $R$ ) invertible matrices $P,Q$ such that $A=PBQ$ .

Lemma 1.3 ([Coh95, Corollary 6.3.6]).

A linear square matrix over $\mathbb{K}\langle X\rangle$ which is not full is associated over $\mathbb{K}$ to a linear hollow matrix.

Remark. A hollow square matrix cannot be full [Coh85, Section 3.2], illustrated in an example:

[TABLE]

Definition 1.4 (Linear Representations, Dimension, Rank [CR94, CR99]).

Let $f\in\mathbb{K}(\!\langle X\rangle\!)$ . A linear representation of $f$ is a triple $\pi_{f}$ = $(u,A,v)$ with $u\in\mathbb{K}^{1\times n}$ , full $A=A_{0}\otimes 1+A_{1}\otimes x_{1}+\ldots+A_{d}\otimes x_{d}$ , $A_{\ell}\in\mathbb{K}^{n\times n}$ and $v\in\mathbb{K}^{n\times 1}$ such that $f=uA^{-1}v$ . The dimension of the representation is $\dim\pi_{f}=n$ . It is called minimal if $A$ has the smallest possible dimension among all linear representations of $f$ . A minimal one $\pi_{f}$ defines the rank of $f$ , $\operatorname{rank}f=\dim\pi_{f}$ . The “empty” representation $\pi=(,,)$ is the minimal one of $0\in\mathbb{K}(\!\langle X\rangle\!)$ with $\dim\pi=0$ .

Definition 1.5 (Left and Right Families

[CR94]).

Let $\pi=(u,A,v)$ be a linear representation of $f\in\mathbb{K}(\!\langle X\rangle\!)$ of dimension $n$ . The families $(s_{1},s_{2},\ldots,s_{n})\subseteq\mathbb{K}(\!\langle X\rangle\!)$ with $s_{i}=(A^{-1}v)_{i}$ and $(t_{1},t_{2},\ldots,t_{n})\subseteq\mathbb{K}(\!\langle X\rangle\!)$ with $t_{j}=(uA^{-1})_{j}$ are called left family and right family respectively. $L(\pi)=\operatorname{span}\{s_{1},s_{2},\ldots,s_{n}\}$ and $R(\pi)=\operatorname{span}\{t_{1},t_{2},\ldots,t_{n}\}$ denote their linear spans (over $\mathbb{K}$ ).

Proposition 1.6 ([CR94], Proposition 4.7).

A representation $\pi=(u,A,v)$ of an element $f\in\mathbb{K}(\!\langle X\rangle\!)$ is minimal if and only if both, the left family and the right family are $\mathbb{K}$ -linearly independent.

Definition 1.7 (Admissible Linear Systems [Coh72]. Admissible Transformations).

A linear representation $\mathcal{A}=(u,A,v)$ of $f\in\mathbb{K}(\!\langle X\rangle\!)$ is called admissible linear system (for $f$ ), denoted by $As=v$ , if $u=e_{1}=[1,0,\ldots,0]$ . The element $f$ is then the first component of the (unique) solution vector $s$ . Given a linear representation $\mathcal{A}=(u,A,v)$ of dimension $n$ of $f\in\mathbb{K}(\!\langle X\rangle\!)$ and invertible matrices $P,Q\in\mathbb{K}^{n\times n}$ , the transformed $P\mathcal{A}Q=(uQ,PAQ,Pv)$ is again a linear representation (of $f$ ). If $\mathcal{A}$ is an ALS, the transformation $(P,Q)$ is called admissible if the first row of $Q$ is $e_{1}=[1,0,\ldots,0]$ .

Transformations can be done by elementary row- and column operations, explained in detail in [Sch17, Remark 1.12]. For further remarks and connections to the related concepts of linearization and realization see [Sch17, Section 1]. For rational operations on ALS level see the following proposition. If an ALS is minimal then more refined versions of an inverse give a minimal ALS again. For a detailled discussion we refer to [Sch17, Section 4].

Proposition 1.8 (Rational Operations [CR99]).

Let $0\neq f,g\in\mathbb{K}(\!\langle X\rangle\!)$ be given by the admissible linear systems $\mathcal{A}_{f}=(u_{f},A_{f},v_{f})$ and $\mathcal{A}_{g}=(u_{g},A_{g},v_{g})$ respectively and let $0\neq\mu\in\mathbb{K}$ . Then admissible linear systems for the rational operations can be obtained as follows:

The scalar multiplication $\mu f$ is given by

[TABLE]

The sum $f+g$ is given by

[TABLE]

The product $fg$ is given by

[TABLE]

And the inverse $f^{-1}$ is given by

[TABLE]

Definition 1.9.

An element in $\mathbb{K}(\!\langle X\rangle\!)$ is called regular, if it has a linear representation $(u,A,v)$ with $A=I-M$ , where $M=M_{1}\otimes x_{1}+\ldots+M_{d}\otimes x_{d}$ with $M_{i}\in\mathbb{K}^{n\times n}$ for some $n\in\mathbb{N}$ , that is, $A_{0}=I$ in Definition 1.4, or equivalently, if $A_{0}$ is regular (invertible).

Remark. $A=I-M$ is also called a monic pencil [HV07]. A regular element can also be represented by a proper linear system $s=v+Ms$ [SS78, Section II.1].

Remark. For a polynomial $p\in\mathbb{K}\langle X\rangle\subseteq\mathbb{K}^{\text{rat}}\langle\!\langle X\rangle\!\rangle\subseteq\mathbb{K}(\!\langle X\rangle\!)$ the rank of $p$ is just the Hankel rank, that is, the rank of its Hankel matrix $\mathcal{H}(p)=(h_{w_{1},w_{2}})$ —rows and columns are indexed by words in the free monoid— where $h_{w_{1},w_{2}}\in\mathbb{K}$ is the coefficient of the monomial $w=w_{1}w_{2}$ of $p$ . See [Fli74] and [SS78, Section II.3].

Example. The Hankel matrix for $p=x(1-yx)=x-xyx$ , with row indices $[1,x,xy,xyx]$ and column indices $[1,x,yx,xyx]$ , is —without zero rows $\{y,x^{2},yx,y^{2},$ $x^{3},x^{2}y,yx^{2},\ldots\}$ and columns $\{y,x^{2},xy,y^{2},x^{3},x^{2}y,yx^{2},\ldots\}$ —

[TABLE]

Its rank is $4$ . Thus $\operatorname{rank}p=4$ and therefore the dimension of any minimal admissible linear system is 4, as will be seen later in the ALS (2.8) in Example 2.7, where we show minimality by Proposition 1.6.

The following definitions follow mainly [BS15] and are streamlined to our purpose. We do not need the full generality here. While there is a rather uniform factorization theory in the commutative setting [GHK06, Section 1.1], even the “simplest” non-commutative case, that is, a “unique” factorization domain like the free associative algebra, is not straightforward. For a general (algebraic) point of view we recommend the survey [Sme15]. Factorization in free ideal rings (FIRs) is discussed in detail in [Coh85, Chapter 3]. FIRs play an important role in the construction of free fields. More on “non-commutative” factorization can be found in [Jor89] and [BHL17] (just to mention a few) and the literature therein.

Definition 1.10 (Similar Right Ideals, Similar Elements

[Coh85, Section 3.2]).

Let $R$ be a ring. Two right ideals $\mathfrak{a},\mathfrak{b}\subseteq R$ are called similar, written as $\mathfrak{a}\sim\mathfrak{b}$ , if $R/\mathfrak{a}\cong R/\mathfrak{b}$ as right $R$ -modules. Two elements $p,q\in R$ are called similar if their right ideals $pR$ and $qR$ are similar, that is, $pR\sim qR$ . See also [Sme15, Section 4.1].

Definition 1.11 (Left and Right Coprime Elements

[BS15, Section 2]).

Let $R$ be a domain and $H=R^{\bullet}=R\setminus\{0\}$ . An element $p\in H$ left divides $q\in H$ , written as $p\!\mid_{\mkern-1.0mu\text{l}}\!q$ , if $q\in pH=\{ph\mid h\in H\}$ . Two elements $p,q$ are called left coprime if for all $h$ such that $h\!\mid_{\mkern-1.0mu\text{l}}\!p$ and $h\!\mid_{\mkern-1.0mu\text{l}}\!q$ implies $h\in H^{\times}=\{f\in H\mid f\text{ is invertible}\}$ , that is, $h$ is an element of the group of units. Right division $p\!\mid_{\mkern-1.0mu\text{r}}\!q$ and the notion of right coprime is defined in a similar way. Two elements are called coprime if they are left and right coprime.

Definition 1.12 (Atomic Domains, Irreducible Elements

[BS15, Section 2]).

Let $R$ be a domain and $H=R^{\bullet}$ . An element $p\in H\setminus H^{\times}$ , that is, a non-zero non-unit (in $R$ ), is called an atom (or irreducible) if $p=q_{1}q_{2}$ with $q_{1},q_{2}\in H$ implies that either $q_{1}\in H^{\times}$ or $q_{2}\in H^{\times}$ . The set of atoms in $R$ is denoted by $\mathbf{A}(R)$ . The (cancellative) monoid $H$ is called atomic if every non-unit can be written as a finite product of atoms of $H$ . The domain $R$ is called atomic if the monoid $R^{\bullet}$ is atomic.

Remark. Similarity of two elements $a,a^{\prime}$ in a weak Bezout ring $R$ is equivalent to the existence of $b,b^{\prime}\in R$ such that $ab^{\prime}=ba^{\prime}$ with $ab^{\prime}$ and $ba^{\prime}$ coprime, that is, $a$ and $b$ are left coprime and $b^{\prime}$ and $a^{\prime}$ are right coprime. The free associative algebra $\mathbb{K}\langle X\rangle$ is a weak Bezout ring [Coh63, Proposition 5.3 and Theorem 6.2].

Example. The polynomials $p=(1-xy)$ and $q=(1-yx)$ are similar, because $px=(1-xy)x=x-xyx=x(1-yx)=xq$ . See also Example 2.7.

Example. In the free monoid $X^{*}$ the atoms are just the letters $x_{i}\in X$ .

Definition 1.13 (Similarity Unique Factorization Domains

[Sme15, Definition 4.1]).

A domain $R$ is called similarity factorial (or a similarity-UFD) if $R$ is atomic and it satisfies the property that if $p_{1}p_{2}\cdots p_{m}=q_{1}q_{2}\cdots q_{n}$ for atoms (irreducible elements) $p_{i},q_{j}\in R$ , then $m=n$ and there exists a permutation $\sigma\in\mathfrak{S}_{m}$ such that $p_{i}$ is similar to $q_{\sigma(i)}$ for all $i\in 1,2,\ldots,m$ .

Proposition 1.14 ([Coh63, Theorem 6.3]).

The free associative algebra $\mathbb{K}\langle X\rangle$ is a similarity (unique) factorization domain.

2 Factorizing non-commutative Polynomials

Our concept for the factorization of nc polynomials (Theorem 2.24) relies on minimal linear representations. Beside the “classical” algorithms from [CC80] and [FM80], there is a naive one illustrated in Section 4. The latter is not very efficient in general for an alphabet with more than one letter but it preserves the form defined in the following. After formulating a minimal polynomial multiplication in Proposition 2.5 and illustrating it in an example we present an algorithm which works directly on the system matrix (of the admissible linear system).

Using proper linear systems [SS78, Section II.1] would be slightly too restrictive because the only possible admissible transformations are conjugations (with respect to the system matrix). That these are not sufficient can be seen in Example 2.22. On the other hand, admissible linear systems are too general. Therefore we define a form that is suitable for our purpose.

Remark. Although we use admissible linear systems, restricting the application of the rational operations (excluding the inverse) in Proposition 1.8 to (systems for) polynomials only, one gets again polynomials. If the inverse is restricted to (systems for) rational formal power series with non-vanishing constant coefficient one gets again rational formal power series. Since we are using multiplication only, Theorem 2.24 does not rely on the (construction of the) free field.

Definition 2.1 (Pre-Standard ALS, Pre-Standard Admissible Transformation).

An admissible linear system $\mathcal{A}=(u,A,v)$ of dimension $n$ with system matrix $A=(a_{ij})$ for a non-zero polynomial $0\neq p\in\mathbb{K}\langle X\rangle$ is called pre-standard, if

(1)

$v=[0,\ldots,0,\lambda]^{\!\top}$ for some $\lambda\in\mathbb{K}$ and

(2)

$a_{ii}=1$ for $i=1,2,\ldots,n$ and $a_{ij}=0$ for $i>j$ , that is, $A$ is upper triangular.

A pre-standard ALS is also written as $\mathcal{A}=(1,A,\lambda)$ with $1,\lambda\in\mathbb{K}$ . An admissible transformation $(P,Q)$ for an ALS $\mathcal{A}$ is called pre-standard, if the transformed system $P\mathcal{A}Q$ is (again) pre-standard.

2.1 Minimal Multiplication

To be able to prove that the construction in Proposition 2.5 leads to a minimal linear representation (for the product of two nc polynomials) some preparation is necessary. One of the main tools is Lemma 1.3 [Coh95, Corollary 6.3.6]. Although we are working with regular elements only, invertibility of the constant coefficient matrix $A_{0}$ (in the system matrix) does not have to be assumed in Lemma 2.2.

Lemma 2.2.

Let $\mathcal{A}=(u,A,v)$ be an ALS of dimension $n\geq 1$ with $\mathbb{K}$ -linearly independent left family $s=A^{-1}v$ and $B=B_{0}\otimes 1+B_{1}\otimes x_{1}+\ldots+B_{d}\otimes x_{d}$ with $B_{\ell}\in\mathbb{K}^{m\times n}$ , such that $Bs=0$ . Then there exists a (unique) $T\in\mathbb{K}^{m\times n}$ such that $B=TA$ .

Proof.

The trivial case $n=1$ implies $B=0$ and therefore $T=0$ . Now let $n\geq 2$ . Without loss of generality assume that $v=[0,\ldots,0,1]^{\!\top}$ and $m=1$ . Since $A$ is full and thus invertible over the free field, there exists a unique $T$ such that $B=TA$ , namely $T=BA^{-1}$ in $\mathbb{K}(\!\langle X\rangle\!)^{1\times n}$ . The last column in $T$ is zero because $0=Bs=BA^{-1}v=Tv$ . Now let $A^{\prime}$ denote the matrix $A$ whose last row is removed and $A^{\prime}_{B}$ the matrix obtained from $A$ when the last row is replaced by $B$ . $A^{\prime}_{B}$ cannot be full since $s\in\ker A^{\prime}_{B}$ would give a contradiction: $s=(A^{\prime}_{B})^{-1}0=0$ .

We claim that there is only one possibility to transform $A^{\prime}_{B}$ to a hollow matrix, namely with zero last row. If we cannot produce a $(n-i)\times i$ block of zeros (by invertible transformations) in the first $n-1$ rows of $A^{\prime}_{B}$ , then we cannot get blocks of zeros of size $(n-i+1)\times i$ and we are done.

Now assume to the contrary that there are invertible matrices $P^{\prime}\in\mathbb{K}^{(n-1)\times(n-1)}$ and (admissible) $Q\in\mathbb{K}^{n\times n}$ with $(Q^{-1}s)_{1}=s_{1}$ , such that $P^{\prime}A^{\prime}Q$ contains a zero block of size $(n-i)\times i$ for some $i=1,\ldots,n-1$ . There are two cases. If the first $n-i$ entries in column 1 cannot be made zero, we construct an upper right zero block:

[TABLE]

where $A_{11}$ has size $(n-i)\times(n-i)$ . If $A_{11}$ were not full, then $A$ would not be full (the last row is not involved in the transformation). Hence this pivot block is invertible over the free field. Therefore $\hat{s}_{1}=\hat{s}_{2}=\ldots=\hat{s}_{n-i}=0$ . Otherwise we construct an upper left zero block in $PAQ$ . But then $\hat{s}_{i+1}=\hat{s}_{i+2}=\ldots=\hat{s}_{n}=0$ . Both contradict $\mathbb{K}$ -linear independence of the left family.

Hence, by Lemma 1.3, $A^{\prime}_{B}$ is associated over $\mathbb{K}$ to a linear hollow matrix with a $1\times n$ block of zeros, say in the last row (columns are left untouched):

[TABLE]

The matrix $T=[-T^{\prime},0]\in\mathbb{K}^{1\times n}$ satisfies $B=TA$ . ∎

Remark. Although the ALS in Lemma 2.2 does not have to be minimal, $\mathbb{K}$ -linear independence of the left family is an important assumption for two reasons. One is connected to “pathological” situations, compare with [Sch17, Example 2.5]. An entry corresponding to some $s_{j}=0$ , say for $j=3$ , could be arbitrary:

[TABLE]

For $B=[2,-2x,y]$ the transformation $T$ has non-scalar entries: $T=[2,yz^{-1},0]$ . The other reason concerns the exclusion of other possibilities for non-fullness except the last row. For $B=[0,0,1]$ the matrix

[TABLE]

is hollow. However, the transformation we are looking for here is $T=[0,z^{-1},0]$ .

Lemma 2.3.

Let $\mathcal{A}=(u,A,v)$ be a pre-standard ALS of dimension $n\geq 2$ with $\mathbb{K}$ -linearly dependent left family $s=A^{-1}v$ . Let $A=(a_{ij})$ . Let $m\in\{2,3,\ldots,n\}$ be the minimal index such that the left subfamily $\underline{s}=(A^{-1}v)_{i=m}^{n}$ is $\mathbb{K}$ -linearly independent. Then there exist matrices $T,U\in\mathbb{K}^{1\times(n+1-m)}$ such that

[TABLE]

Proof.

By assumption, the left subfamily $(s_{m-1},s_{m},\ldots,s_{n})$ is $\mathbb{K}$ -linearly dependent. Thus there are $\kappa_{j}\in\mathbb{K}$ such that $s_{m-1}=\kappa_{m}s_{m}+\kappa_{m+1}s_{m+1}+\ldots+\kappa_{n}s_{n}$ . Let $U=[\kappa_{m},\kappa_{m+1},\ldots,\kappa_{n}]$ . Then $s_{m-1}-U\underline{s}=0$ . Since $\mathcal{A}$ is pre-standard, $v_{m-1}=0$ . Now we can apply Lemma 2.2 with $B=U+[a_{m-1,m},a_{m-1,m+1},\ldots,a_{m-1,n}]$ (and $\underline{s}$ ). Hence, there exists a matrix $T\in\mathbb{K}^{1\times(n+1-m)}$ such that

[TABLE]

Recall that the last column of $T$ is zero, whence $T(v_{i})_{i=m}^{n}=0$ . ∎

Proposition 2.5 (Minimal Polynomial Multiplication).

Let $0\neq p,q\in\mathbb{K}\langle X\rangle$ be given by the minimal pre-standard admissible linear systems $A_{p}=(u_{p},A_{p},v_{p})=(1,A_{p},\lambda_{p})$ and $A_{q}=(u_{q},A_{q},v_{q})=(1,A_{q},\lambda_{q})$ of dimension $n_{p},n_{q}\geq 2$ respectively. Then a minimal ALS for $pq$ has dimension $n=n_{p}+n_{q}-1$ and can be constructed in the following way:

(1)

Construct the following ALS $\mathcal{A}^{\prime}=(u^{\prime},A^{\prime},v^{\prime})$ for the product $pq$ :

[TABLE]

(2)

Add $\lambda_{p}$ -times column $n_{p}$ to column $n_{p}+1$ (the entry $s_{n_{p}}^{\prime}$ becomes zero).

(3)

Remove row $n_{p}$ of $A^{\prime}$ and $v^{\prime}$ and column $n_{p}$ of $A^{\prime}$ and $u^{\prime}$ and denote the new (pre-standard) ALS of dimension $n_{p}+n_{q}-1$ by $\mathcal{A}=(u,A,v)=(1,A,\lambda)$ .

Proof.

The left family of $\mathcal{A}^{\prime}$ is

[TABLE]

Clearly, $\mathcal{A}$ is again pre-standard with $\lambda=\lambda_{q}$ . Both systems $\mathcal{A}_{p}$ and $\mathcal{A}_{q}$ are minimal. Therefore their left and right families are $\mathbb{K}$ -linearly independent. Without loss of generality assume that $\lambda_{p}=1$ . Then the last entry $t_{n_{p}}^{p}$ of the right family of $\mathcal{A}_{p}$ is equal to $p$ . Let $k=n_{p}$ . We have to show that both, the left and the right family

[TABLE]

of $\mathcal{A}$ are $\mathbb{K}$ -linearly independent respectively. Assume the contrary for $s$ , say there is an index $1<m\leq k$ such that $(s_{m-1},s_{m},\ldots,s_{n})$ is $\mathbb{K}$ -linearly dependent while $(s_{m},\ldots,s_{n})$ is $\mathbb{K}$ -linearly independent. Then, by Lemma 2.3 there exist matrices $T,U\in\mathbb{K}^{1\times(n-m+1)}$ such that $\eqref{eqn:pf.min1}$ holds and therefore invertible matrices $P,Q\in\mathbb{K}^{n\times n}$ ,

[TABLE]

that yield equation $s_{m-1}=0$ (in row $m-1$ ) in $P\mathcal{A}Q$ . Let $\tilde{P}$ (respectively $\tilde{Q}$ ) be the upper left part of $P$ (respectively $Q$ ) of size $k\times k$ . Then the equation in row $m-1$ in $\tilde{P}\mathcal{A}_{p}\tilde{Q}$ is $s^{p}_{m-1}=\alpha\in\mathbb{K}$ , contradicting $\mathbb{K}$ -linear independence of the left family of $\mathcal{A}_{p}$ since $s^{p}_{k}=\lambda_{p}\in\mathbb{K}$ . A similar argument (and a variant of Lemma 2.3) for the right family yields its $\mathbb{K}$ -linear independence. Hence, by Proposition 1.6, the admissible linear system $\mathcal{A}$ (for $pq$ ) is minimal. ∎

Corollary 2.6.

Let $0\neq p,q\in\mathbb{K}\langle X\rangle$ . Then $\operatorname{rank}(pq)=\operatorname{rank}(p)+\operatorname{rank}(q)-1$ .

Example 2.7.

The polynomials $p=x\in\mathbb{K}\langle X\rangle$ and $q=1-yx\in\mathbb{K}\langle X\rangle$ have the minimal pre-standard admissible linear systems

[TABLE]

respectively. Then a pre-standard ALS for $pq=x(1-yx)$ is given by

[TABLE]

Adding column 2 to column 3 (and subtracting $s_{3}$ from $s_{2}$ ) yields

[TABLE]

thus the pre-standard ALS

[TABLE]

Since also the right family $t=[1,x,-xy,x(1-yx)]$ is $\mathbb{K}$ -linearly independent, this system is minimal by Proposition 1.6. Note the upper right $1\times 2$ block of zeros in the system matrix of $\mathcal{A}$ .

2.2 Minimizing a pre-standard ALS

A close look on the proof of Proposition 2.5 reveals a surprisingly simple algorithm for the construction of a minimal pre-standard admissible linear system, provided that one in pre-standard form is given. It can be used for minimizing the ALS for the sum in Proposition 1.8. “Simple” means that it can easily be done manually also for quite large sparse systems. Depending on the data structure, the implementation itself is somewhat technical. One has to be very careful if the scalars (from the ground field $\mathbb{K}$ ) cannot be represented exactly, especially when systems of linear equations (see below) have to be solved.

To illustrate the main idea we (partially) minimize a non-minimal “almost” pre-standard ALS $\mathcal{A}=(u,A,v)$ of dimension $n=6$ for $p=-xy+(xy+z)$ . Note that we do not need knowledge of the left and right family at all. Let

[TABLE]

First we do one “left” minimization step, that is, we remove (if possible) one element of the $\mathbb{K}$ -linearly dependent left family $s=A^{-1}v$ and construct a new system. We fix a $1\leq k<n$ , say $k=3$ . If we find a (pre-standard admissible) transformation $(P,Q)$ of the form

[TABLE]

such that row $k$ in $PAQ$ is $[0,0,1,0,0,0]$ and $(Pv)_{k}=0$ , we can eliminate row $k$ and column $k$ in $P\mathcal{A}Q$ because $(Q^{-1}s)_{k}=0$ . How can we find these blocks $T,U\in\mathbb{K}^{1\times(n-k)}$ ? We write $\mathcal{A}$ in block form —block row and column indices are underlined to distinguish them from component indices— (with respect to row/column $k$ )

[TABLE]

and apply the transformation $(P,Q)$ :

[TABLE]

Now we can read of a sufficient condition for $(Q^{-1}s)_{k}=0$ , namely the existence of $T,U\in\mathbb{K}^{1\times(n-k)}$ such that

[TABLE]

Let $d$ be the number of letters in our alphabet $X$ . The blocks $T=[\alpha_{k+1},\alpha_{k+2},\ldots,\alpha_{n}]$ and $U=[\beta_{k+1},\beta_{k+2},\ldots,\beta_{n}]$ in the transformation $(P,Q)$ are of size $1\times(n-k)$ , thus we have a linear system of equations (over $\mathbb{K}$ ) with $2(n-k)$ unknowns (for $k>1$ ) and $(d+1)(n-k)+1$ equations:

[TABLE]

One solution is $T=[0,0,1]$ and $U=[0,0,-1]$ . We compute $\tilde{\mathcal{A}}_{1}=P\mathcal{A}Q$ and remove block row $\underline{2}$ and column $\underline{2}$ , that is, row $k$ and column $k$ , to get the new ALS

[TABLE]

Next we do one “right” minimization step, that is, we remove (if possible) one element of the $\mathbb{K}$ -linearly dependent right family $t=uA^{-1}$ and construct a new system. We fix a $1<k\leq n=5$ , say $k=3$ . Note that $t_{1}=1$ . Now we are looking for a transformation $(P,Q)$ of the form

[TABLE]

such that column $k$ in $PAQ$ is $[0,0,1,0,0]^{\!\top}$ (for an admissible transformation, that is, $U$ has a zero first row, the corresponding entry $u_{k}$ in $u$ is zero). A sufficient condition for $(tP^{-1})_{k}=0$ is the existence of $T,U\in\mathbb{K}^{(k-1)\times 1}$ such that

[TABLE]

(In Remark 2.18 there is a less “compressed” version of this linear system of equations.) One solution is $T=[1,0]^{\!\top}$ and $U=[0,0]^{\!\top}$ . We compute $\tilde{\mathcal{A}}_{2}=P\mathcal{A}_{1}Q$ and remove row $k$ and column $k$ to get the new (not yet minimal) ALS

[TABLE]

If a left (respectively right) minimization step with $k=1$ (respectively $k=n$ and $v=[0,\ldots,0,\lambda]^{\!\top}$ ) can be done, then the ALS represents zero and we can stop immediately.

The following is the only non-trival observation: Recall that, if there exist row (respectively column) blocks $T,U$ such that (2.12) (respectively (2.14)) has a solution then the left (respectively right) family is $\mathbb{K}$ -linearly dependent. To guarantee minimality by Proposition 1.6 we need the other implication, that is, the existence of appropriate row or column blocks for non-minimal pre-standard admissible linear systems.

Although the arguments can be found in the proof of Proposition 2.5, we repeat them here because this is the crucial part of the minimization algorithm: Let $\mathcal{A}=(u,A,v)$ be a pre-standard ALS of dimension $n\geq 2$ with left family $s=(s_{1},s_{2},\ldots,s_{n})$ and assume that there exists a $1\leq k<n$ such that the subfamily $(s_{k+1},s_{k+2},\ldots,s_{n})$ is $\mathbb{K}$ -linearly independent while $(s_{k},s_{k+1},\ldots,s_{n})$ is $\mathbb{K}$ -linearly dependent. Then, by Lemma 2.3, there exist matrices $T,U\in\mathbb{K}^{1\times(n-k)}$ such that (2.12) holds. In other words: We have to start with $k_{s}=n-1$ for a left and $k_{t}=2$ for a right minimization step.

If we apply one minimization step, we must check the other family “again”, illustrated in the following example, which is not constructed out of two minimal systems:

[TABLE]

Clearly, the left subfamily $(s_{3},s_{4},s_{5})$ and the right subfamily $(t_{1},t_{2},t_{3})$ of $\mathcal{A}$ are $\mathbb{K}$ -linearly independent respectively. If we subtract row 3 from row 2 and add column 2 to column 3, we get the ALS

[TABLE]

The right subfamily $(t_{1}^{\prime\prime},t_{2}^{\prime\prime},t_{3}^{\prime\prime})$ of $\mathcal{A}^{\prime\prime}=\mathcal{A}^{\prime}\rule[-2.15277pt]{0.0pt}{9.47217pt}^{[-2]}$ is (here) not $\mathbb{K}$ -linearly independent anymore, therefore we must check for a right minimization step for $k=3$ again.

Definition 2.15 (Minimization Equations, Minimization Transformations).

Let $\mathcal{A}=(u,A,v)$ be a pre-standard ALS of dimension $n\geq 2$ . Recall the block decomposition (2.11)

[TABLE]

By $\mathcal{A}^{[-k]}$ we denote the ALS $\mathcal{A}^{[k]}$ without row/column $k$ (of dimension $n-1$ ):

[TABLE]

For $k=\{1,2,\ldots,n-1\}$ the equations $U+A_{2,3}+TA_{3,3}=0$ and $v_{\underline{2}}+Tv_{\underline{3}}=0$ , see (2.12), with respect to the block decomposition $\mathcal{A}^{[k]}$ are called left minimization equations, denoted by $\mathcal{L}_{k}=\mathcal{L}_{k}(\mathcal{A})$ . A solution by the row block pair $(T,U)$ is denoted by $\mathcal{L}_{k}(T,U)=0$ , the corresponding transformation $(P,Q)=\bigl{(}P(T),Q(U)\bigr{)}$ , see (2.10), is called left minimization transformation. For $k=\{2,3,\ldots,n\}$ the equations $A_{1,1}U+A_{1,2}+T=0$ , see (2.14), with respect to the block decomposition $\mathcal{A}^{[k]}$ are called right minimization equations, denoted by $\mathcal{R}_{k}=\mathcal{R}_{k}(\mathcal{A})$ . A solution by the column block pair $(T,U)$ is denoted by $\mathcal{R}_{k}(T,U)=0$ , the corresponding transformation $(P,Q)=\bigl{(}P(T),Q(U)\bigr{)}$ , see (2.13), is called right minimization transformation.

Now there is only one important detail left, namely that we cannot apply Lemma 2.3 in the following (first) left minimization step. For $0\neq\alpha\in\mathbb{K}$ consider the ALS $\mathcal{A}$

[TABLE]

and (admissibly) transform the system matrix $A$ in the following way:

[TABLE]

thus

[TABLE]

and we can remove the last row and column. Note that we do not have to consider such a special case for the right family.

Algorithm 2.17 (Minimizing a pre-standard ALS).

*Input: $\mathcal{A}=(u,A,v)$ pre-standard ALS of dimension $n\geq 2$ (for some polynomial $p$ ).

Output: $\mathcal{A}^{\prime}=(,,)$ if $p=0$ or a minimal pre-standard ALS $\mathcal{A}^{\prime}=(u^{\prime},A^{\prime},v^{\prime})$ if $p\neq 0$ .*

[TABLE]

Proof.

The admissible linear system $\mathcal{A}$ represents $p=0$ if and only if $s_{1}=(A^{-1}v)_{1}=0$ . Since all systems are equivalent to $\mathcal{A}$ , this case is recognized for $k^{\prime}=1$ because by Lemma 2.3 there is an admissible transformation such that the first left minimization equation is fulfilled. Now assume $p\neq 0$ . We have to show that both, the left family $s^{\prime}$ and the right family $t^{\prime}$ of $\mathcal{A}^{\prime}=(u^{\prime},A^{\prime},v^{\prime})$ are $\mathbb{K}$ -linearly independent respectively. Let $n^{\prime}=\dim(\mathcal{A}^{\prime})$ and for $k\in\{1,2,\ldots,n^{\prime}\}$ denote by $s^{\prime}_{(k)}=(s^{\prime}_{n^{\prime}+1-k},s^{\prime}_{n^{\prime}+2-k},\ldots,s^{\prime}_{n^{\prime}})$ the left and by $t^{\prime}_{(k)}=(t^{\prime}_{1},t^{\prime}_{2},\ldots,t^{\prime}_{k})$ the right subfamily. By assumption $\mathcal{A}$ is a pre-standard ALS and therefore $s^{\prime}_{n^{\prime}}\neq 0$ and $t^{\prime}_{1}\neq 0$ , that is, $s^{\prime}_{(1)}$ is $\mathbb{K}$ -linearly independent and $t^{\prime}_{(1)}$ is $\mathbb{K}$ -linearly independent. The loop starts with $k=2$ . Only if both, $s^{\prime}_{(k)}$ and $t^{\prime}_{(k)}$ are $\mathbb{K}$ -linearly indpendent respectively, $k$ is incremented. For $k=2$ we can solve the special case by (2.16) for the left family. Otherwise a left (Lemma 2.3) or a right (variant of Lemma 2.3) minimization step was successful and the dimension of the current ALS is strictly smaller than that of the previous. Hence, since $k$ is bounded from below, the algorithm stops in a finite number of steps. We just have to make sure that there exists an admissible transformation, if there exists a column couple $(T,U)$ such that the right minimization equations $\mathcal{R}_{k}(T,U)=0$ are fulfilled. However, if the first column would be involved to eliminate the first entry in column $k$ , the $k$ -th row could be used instead. Clearly, $\mathcal{A}^{\prime}$ is in pre-standard form. ∎

Remark 2.18.

This algorithm can be implemented very efficiently provided that row and column transformations are done directly. Let $d$ be the number of letters in our alphabet $X$ . For $\ell=0,1,\ldots,d$ let $A_{ij}^{(\ell)}$ denote the submatrix corresponding to letter $x_{\ell}$ and the (current) block decomposition of $\mathcal{A}^{[k]}$ . The right minimization equations $A_{1,1}U+A_{1,2}+T=0$ can be written as

[TABLE]

with $2(k-1)$ unknowns, $k<n$ . By Gaussian elimination one gets complexity $\mathcal{O}(dn^{3})$ for solving such a system, see [Dem97, Section 2.3]. To build such a system and working on a linear matrix pencil $\bigl{[}\begin{smallmatrix}0&u\\ v&A\end{smallmatrix}\bigr{]}$ with $d+1$ square coefficient matrices of size $n+1$ (transformations, etc.) has complexity $\mathcal{O}(dn^{2})$ . Since there are at most $2(n-1)$ steps, we get overall (minimization) complexity $\mathcal{O}(dn^{4})$ . The algorithm of [CC80] has complexity $\mathcal{O}(dn^{3})$ . One has to be careful with a direct comparison. The latter works more general for regular elements, that is, rational formal power series. However the idea used here generalizes directly for larger blocks, say for a block decomposition $\mathcal{A}^{[k,k+1,\ldots,k+m]}$ for $k,m<n$ and can be used partially for non-regular elements in the free field, for example to solve the word problem which would have complexity $\mathcal{O}(dn^{6})$ . Further details can be found in [Sch17].

2.3 Factorizing non-commutative Polynomials

Definition 2.19 (Atomic Admissible Linear Systems).

A minimal pre-standard ALS $\mathcal{A}=(1,A,\lambda)$ of dimension $n\geq 2$ is called atomic (irreducible), if there is no pre-standard admissible transformation $(P,Q)$ such that $PAQ$ has an upper right block of zeros of size $(n-i-1)\times i$ for some $i=1,2,\ldots,n-2$ .

If we take the minimal ALS (2.8) for $pq=x(1-yx)$ from Example 2.7 and add column 2 to column 4, we obtain

[TABLE]

Subtracting row 3 from row 1 yields

[TABLE]

with an upper right $2\times 1$ block of zeros in the system matrix. We would obtain the same system by minimal polynomial multiplication of $1-xy$ and $x$ . This illustrates that we can find the factors of non-commutative polynomials by searching for pre-standard admissible transformations that give a corresponding upper right block of zeros in the transformed system matrix.

Theorem 2.24 will establish the correspondence between a factorization into atoms and the structure of the upper right blocks of zeros. Thus, to factorize a polynomial $p$ of rank $n\geq 3$ into non-trivial factors $p=q_{1}q_{2}$ with $\operatorname{rank}(q_{i})=n_{i}\geq 2$ and $n=n_{1}+n_{2}-1$ , we have to look for (pre-standard admissible) transformations of the form

[TABLE]

with entries $\alpha_{ij},\beta_{ij}\in\mathbb{K}$ . In general this is a non-linear problem with $(n-2)(n-1)$ unknowns.

Definition 2.21 (Standard Admissible Linear Systems).

Every

•

ALS $(1,I_{1},\lambda)$ for a scalar $0\neq\lambda\in\mathbb{K}$ ,

•

atomic ALS and

•

non-atomic minimal pre-standard ALS $\mathcal{A}=(1,A,\lambda)$ for $p\in\mathbb{K}\langle X\rangle$ of dimension $n\geq 3$ obtained from minimal multiplication (Proposition 2.5) of atomic admissible linear systems for its atomic factors $p=q_{1}q_{2}\cdots q_{m}$ (with atoms $q_{i}$ )

is called standard.

Remark. We have the following chain of “inclusions” of admissible linear systems: Atomic ALS $\subsetneq$ Standard ALS $\subsetneq$ Pre-Standard ALS $\subsetneq$ ALS.

Example 2.22.

Let $p=x^{2}-2\in\mathbb{K}\langle X\rangle$ be given by the minimal ALS

[TABLE]

If $\mathbb{K}=\mathbb{Q}$ then $\mathcal{A}$ is atomic (irreducible). If $\mathbb{K}=\mathbb{R}$ then there is the pre-standard admissible transformation

[TABLE]

such that $\mathcal{A}^{\prime}=P\mathcal{A}Q$ is

[TABLE]

thus $p=x^{2}-2=(x-\sqrt{2})(x+\sqrt{2})$ in $\mathbb{R}\langle X\rangle$ .

Remark. Note that it is easy to check that $p=xy-2\in\mathbb{K}\langle X\rangle$ is atomic, because both entries, $\alpha_{1,2}$ in $P$ and $\beta_{2,3}$ in $Q$ have to be zero (otherwise the upper right entry in $A^{\prime}$ could not become zero) and hence there is no non-trivial pre-standard admissible transformation, that is, transformation that changes the upper right block structure.

Lemma 2.23.

Let $0\neq p,q_{1},q_{2}\in\mathbb{K}\langle X\rangle$ be given by the minimal pre-standard admissible linear systems $\mathcal{A}=(1,A,\lambda)$ , $\mathcal{A}_{1}=(1,A_{1},\lambda_{1})$ and $\mathcal{A}_{2}=(1,A_{2},\lambda_{2})$ of dimension $n=\operatorname{rank}p$ and $n_{1},n_{2}\geq 2$ respectively with $p=q_{1}q_{2}$ . Then there exists a pre-standard admissible transformation $(P,Q)$ such that $PAQ$ has an upper right block of zeros of size $(n_{1}-1)\times(n_{2}-1)=(n-n_{2})\times(n-n_{1})$ .

Proof.

Let $\mathcal{A}^{\prime}=(1,A^{\prime},\lambda_{2})$ be the minimal ALS from Proposition 2.5 for $q_{1}q_{2}$ . Clearly, we have $\dim\mathcal{A}^{\prime}=n_{1}+n_{2}-1=n=\operatorname{rank}p$ . And $A^{\prime}$ has, by construction, an upper right block of zeros of size $(n_{1}-1)\times(n_{2}-1)$ . Both, $\mathcal{A}$ and $\mathcal{A}^{\prime}$ represent the same element $p$ , thus —by [CR99, Theorem 1.4]— there exists an admissible transformation $(P,Q)$ such that $P\mathcal{A}Q=\mathcal{A}^{\prime}$ . Since $\mathcal{A}^{\prime}$ is pre-standard, $(P,Q)$ is a pre-standard admissible transformation. ∎

Theorem 2.24 (Polynomial Factorization).

Let $p\in\mathbb{K}\langle X\rangle$ be given by the minimal pre-standard admissible linear system $\mathcal{A}=(1,A,\lambda)$ of dimension $n=\operatorname{rank}p\geq 3$ . Then $p$ has a factorization into $p=q_{1}q_{2}$ with $\operatorname{rank}(q_{i})=n_{i}\geq 2$ if and only if there exists a pre-standard admissible transformation $(P,Q)$ such that $PAQ$ has an upper right block of zeros of size $(n_{1}-1)\times(n_{2}-1)$ .

Proof.

If there is such a factorization, then Lemma 2.23 applies. Conversely, if we have such a (pre-standard admissible) transformation for a zero block of size $k_{1}\times k_{2}$ we obtain an ALS in block form ( $p$ is the first entry of $s_{\underline{1}}$ )

[TABLE]

with $A_{1,1}$ of size $k_{1}\times k_{1}$ and $A_{3,3}$ of size $k_{2}\times k_{2}$ , in which we duplicate the entry $s_{k_{1}+1}$ by inserting a “dummy” row (and column) to get the following ALS of size $k_{1}+k_{2}+2=n+1$ :

[TABLE]

According to the construction of the multiplication in Proposition 1.8 we have $p=fg$ in the first component of $s_{\underline{1}}$ , the first block in $s^{\prime}$ , for $f,g\in\mathbb{K}\langle X\rangle$ given by the (pre-standard) admissible linear systems

[TABLE]

of dimension $n_{1}=k_{1}+1$ and $n_{2}=k_{2}+1$ respectively. ∎

This finishes the algorithm. A simple analogue of [CR99, Theorem 4.1] will do the rest, we do not have to worry about invertibility of the transformation matrices $P$ and $Q$ in (2.20). In our case $\mathbb{K}[\alpha,\beta]=\mathbb{K}[\alpha_{1,2},\ldots,\alpha_{1,n-1},\alpha_{2,3},\ldots,\alpha_{2,n-1},\ldots,\alpha_{n-2,n-1},$ $\beta_{2,3},\ldots,\beta_{2,n},$ $\beta_{3,4},\ldots,\beta_{3,n},\ldots,\beta_{n-1,n}]$ . However, a non-trivial ideal does not guarantee a solution over $\mathbb{K}$ although there are solutions over $\overline{\mathbb{K}}$ . To test only (without computing it) if there is a solution (over $\mathbb{K}$ ) one can use the concept of resultants. An introduction can be found in [CLO15, Section 3.6]. This book contains also an introduction to Gröbner bases and an overview about computer algebra software for computing them. Additional to the work of [Buc70], the survey on Gröbner–Shirshov bases of [BK00] could be consulted.

Remark. Note, that in general in order to reverse the multiplication (Proposition 2.5) to find factors, we also need a lower left block of zeros of appropriate size. This important fact is hidden in the pre-standard form of an ALS. A general factorization concept is considered in future work.

Proposition 2.25.

Let $p\in\mathbb{K}\langle X\rangle$ be given by the minimal pre-standard admissible linear system $\mathcal{A}=(1,A,\lambda)$ of dimension $n=\operatorname{rank}p\geq 3$ and let $(P,Q)$ as in (2.20). Fix a $k\in\{1,2,\ldots,n-2\}$ and denote by $I_{k}$ the ideal of $\mathbb{K}[\alpha,\beta]$ generated by the coefficients of each $x\in\{1\}\cup X$ in the $(i,j)$ entries of the matrix $PAQ$ for $1\leq i\leq k$ and $k+2\leq j\leq n$ . Then $p$ factorizes over $\overline{\mathbb{K}}\langle X\rangle$ into $p=q_{1}q_{2}$ with $\operatorname{rank}q_{1}=k+1$ and $\operatorname{rank}q_{2}=n-k$ if and only if the ideal $I_{k}$ is non-trivial.

Given a polynomial $p\in\mathbb{K}\langle X\rangle$ by a minimal pre-standard ALS of dimension $n=\operatorname{rank}p\geq 2$ there are at most $\phi(n)=2^{n-2}$ (minimal) standard admissible linear systems (with respect to the structure of the upper right blocks of zeros). For $n=2$ this is clear. For $n>2$ the ALS could be atomic or have a block of zeros of size $1\times(n-2)$ or $(n-2)\times 1$ , thus $\phi(n+1)=1+2\phi(n)-1=2\phi(n)$ because the system with “finest” upper right structure is counted twice.

Remark. Recall that —up to similarity— each element $p\in\mathbb{K}\langle X\rangle$ has only one factorization into atoms. If one is interested in the number of factorizations (not necessarily irreducible) up to permutation (and multiplication of the factors by units) the above estimate $2^{\operatorname{rank}p-2}$ can be used. However, the number of factorizations in the sense of [BHL17, Definition 3.1] can be bigger. As an example take the polynomial $p=(x-1)(x-2)(x-3)$ which has $3!=6$ different factorizations while the number of standard admissible linear systems is bounded by $2^{\operatorname{rank}p-2}=4$ .

Let $p$ be a non-zero polynomial with the factorization $p=q_{1}q_{2}\cdots q_{m}$ into atoms $q_{i}\in\mathbb{K}\langle X\rangle$ . Since $\mathbb{K}\langle X\rangle$ is a similarity-UFD (Proposition 1.14), each factorization of (a non-zero non-unit) $p$ into atoms has $m$ factors. Therefore one can define the length of $p$ by $m$ , written as $\ell(p)=m$ . For a word $w\in X^{*}\subseteq\mathbb{K}\langle X\rangle$ the length is $\ell(w)=|w|$ . By looking at the minimal polynomial multiplication (Proposition 2.5) it is easy to see that the length of an element $p\in\mathbb{K}\langle X\rangle^{\bullet}$ can be estimated by the rank, namely $\ell(p)\leq\operatorname{rank}(p)-1$ . More on length functions —and transfer homomorphisms in the context of non-unique factorizations— (in the non-commutative setting) can be found in [Sme15, Section 3] or [BS15].

3 Generalizing the Companion Matrix

For a special case, namely an alphabet with just one letter, the companion matrix of a polynomial $p(x)$ yields immediately a minimal linear representation of $p\in\mathbb{K}\langle\{x\}\rangle$ . If this is the characteristic polynomial of some (square) matrix $B\in\mathbb{K}^{m\times m}$ , then its eigenvalues can be computed by the techniques from Section 2 (if necessary going over to $\overline{\mathbb{K}}$ ), illustrated in Example 3.7.

For a broader class of (nc) polynomials, left and right companion systems can be defined. In general minimal pre-standard admissible linear systems are necessary to generalize companion matrices, see Definition 3.6.

Definition 3.1 (Companion Matrix, Characteristic Polynomial, Normal Form

[Gan86, Section 6.6]).

Let $p(x)=a_{0}+a_{1}x+\ldots+a_{m-1}x^{m-1}+x^{m}\in\mathbb{K}[x]$ . The companion matrix $L(p)$ is defined as

[TABLE]

Then $p(x)$ is the characteristic polynomial of $L=L(p)$ :

[TABLE]

Given a square matrix $M\in\mathbb{K}^{m\times m}$ , the normal form for $M$ can be defined in terms of the companion matrix $L(M)$ of its characteristic polynomial $p(M)=\det(xI-M)$ .

Remark. In [Coh95, Section 8.1], $C(p)=xI-L(p)^{\!\top}$ is also called companion matrix. This is justified by viewing $C(p)$ as linear matrix pencil $C(p)=C_{0}\otimes 1+C_{x}\otimes x$ . It generalizes nicely for non-commutative polynomials.

Now we leave $\mathbb{K}[x]=\mathbb{K}\langle\{x\}\rangle$ and consider (nc) polynomials $p\in\mathbb{K}\langle X\rangle$ . There are two cases where a minimal ALS can be stated immediately, namely if the support (of the polynomial) can be “built” from the left (in the left family) or from the right (in the right family) with strictly increasing rank. For example, a minimal pre-standard ALS for $p=a_{0}+a_{1}(x+2y)+a_{2}(x-z)(x+2y)+y(x-z)(x+2y)$ is given by

[TABLE]

Definition 3.2 (Left and Right Companion System).

For $i=1,2,\ldots,m$ let $q_{i}\in\mathbb{K}\langle X\rangle$ with $\operatorname{rank}{q_{i}}=2$ and $a_{i}\in\mathbb{K}$ . For a polynomial $p\in\mathbb{K}\langle X\rangle$ of the form

[TABLE]

the pre-standard ALS

[TABLE]

is called left companion system. And for a polynomial $p\in\mathbb{K}\langle X\rangle$ of the form

[TABLE]

the pre-standard ALS

[TABLE]

is called right companion system.

Proposition 3.5.

For $i=1,2,\ldots,m$ let $q_{i}\in\mathbb{K}\langle X\rangle$ with $\operatorname{rank}{q_{i}}=2$ . Then the polynomials $p_{\text{l}}=q_{m}q_{m-1}\cdots q_{1}+a_{m-1}q_{m-1}\cdots q_{1}+\ldots+a_{2}q_{2}q_{1}+a_{1}q_{1}+a_{0}$ and $p_{\text{r}}=a_{0}+a_{1}q_{1}+a_{2}q_{1}q_{2}+\ldots+a_{m-1}q_{1}q_{2}\cdots q_{m-1}+q_{1}q_{2}\cdots q_{m}$ have rank $m+1$ .

Proof.

Both, the left family $(p,q_{m-1}\cdots q_{1},\ldots,q_{2}q_{1},q_{1},1)$ and the right family $(1,q_{m}+a_{m-1},$ $(q_{m}+a_{m-1})q_{m-1}+a_{m-2},\ldots,p)$ for (3.3) are $\mathbb{K}$ -linearly independent. Thus, the left companion system (for $p_{\text{l}}$ ) is minimal of dimension $m+1$ . Hence $\operatorname{rank}p_{\text{l}}=m+1$ . By a similar argument for the right companion system we get $\operatorname{rank}p_{\text{r}}=m+1$ . ∎

For a general polynomial $p\in\mathbb{K}\langle X\rangle$ with $\operatorname{rank}p=n\geq 2$ we can take any minimal pre-standard ALS $\mathcal{A}=(1,A,\lambda)$ to obtain an ALS of the form $(1,A^{\prime},1)$ by dividing the last row by $\lambda$ and multiplying the last column by $\lambda$ . Now we can define a (generalized version of the) companion matrix. Those companion matrices can be used as building blocks to get companion matrices for products of polynomials. This is just a different point of view on the “minimal” polynomial multiplication from Proposition 2.5.

Remark. Although nothing can be said in general about minimality of a linear representation for commutative polynomials (in several variables), Proposition 3.5 can be used for constructing minimal linear representations in the commutative case. Because in this case, the rank is the maximum of the ranks of the monomials. For example $p=x^{2}y+xyz=xyx+xyz=xy(x+z)$ .

Definition 3.6 (Companion Matrices).

Let $p\in\mathbb{K}\langle X\rangle$ with $\operatorname{rank}p=n\geq 2$ be given by a minimal pre-standard admissible linear system $\mathcal{A}=(1,A,1)$ and denote by $C(p)$ the upper right submatrix of size $(n-1)\times(n-1)$ . Then $C(p)$ is called a (nc) companion matrix of $p$ .

Example 3.7.

Let

[TABLE]

Then the characteristic polynomial of $B$ is $p(x)=\det(xI-B)=x^{3}-10x^{2}+31x-30$ . The left companion system of $p$ is

[TABLE]

Applying the transformation $(P,Q)$ with

[TABLE]

and

[TABLE]

we get the ALS

[TABLE]

Thus the eigenvalues of $B$ are $2$ , $3$ and $5$ . Compare with Example 2.22 for the case when the polynomial does not decompose in linear factors.

4 An Example (step by step)

Let $p=x(1-yx)(3-yx)$ and $q=(xy-1)(xy-3)x$ be given. Taking companion systems (Definition 3.2) for their factors respectively, by Proposition 2.5 we get the minimal ALS (for $p$ ):

[TABLE]

Clearly, $p=xyxyx-4xyx+3x=q$ and $\operatorname{rank}p=\operatorname{rank}q=6$ .

4.1 Constructing a minimal ALS

If a minimal ALS (for some polynomial) cannot be stated directly by a left or right companion system (Definition 3.2), an ALS for $p$ can be constructed using rational operations from Proposition 1.8 (except the inverse) out of monomials (or polynomials). If the involved systems are pre-standard, then the resulting system for the sum can be easily transformed into a pre-standard ALS by row operations and we can apply Algorithm 2.17 to obtain a minimal one.

Here we describe an alternative approach which works directly on the left and right families like [CC80]. It preserves the upper triangular form of the system matrix. However, for an alphabet with more than one letter, it is far from optimal because then the dimension of the vector spaces (see below) might grow exponentially. We leave it to the reader to use the ideas of [CC80] to improve that significantely.

Recall that an ALS $\mathcal{A}=(u,A,v)$ is minimal if and only if both the left family $s=A^{-1}v$ is $\mathbb{K}$ -linearly independent and the right family $t=uA^{-1}$ is $\mathbb{K}$ -linearly independent (Proposition 1.6). Let $f=3x-4xyx$ . An ALS for $f$ of dimension $n=6$ is given by

[TABLE]

Note that the free associative algebra $\mathbb{K}\langle X\rangle$ is a vector space with basis $X^{*}$ , the free monoid of the alphabet $X$ . Here $X^{*}=\{1,x,y,xx,xy,yx,yy,\ldots\}$ . Therefore, we can write the left family $s$ as a matrix (of coordinate row vectors) $S\in\mathbb{K}^{n\times m_{s}}$ with column indices $\{1,x,yx,xyx\}$ and the right family $t$ as a matrix (of coordinate column vectors) $T\in\mathbb{K}^{m_{t}\times n}$ with row indices $\{1,x,xy,xyx\}$ , that is,

[TABLE]

Both, $S$ and $T$ , have rank $4<n=6$ , hence $\mathcal{A}$ cannot be minimal. The entries of the left (respectively right) family are just rows in $S$ , also denoted by $s_{i}$ for $i=1,\ldots,n$ (respectively columns in $T$ , denoted by $t_{j}$ for $j=1,\ldots,n$ ).

Adding a row $s_{k}$ to $s_{i}$ results in subtracting column $i$ from column $k$ in the system matrix $A$ . In order to keep the triangular structure $i<k$ must hold. Similarly, adding a column $t_{k}$ to $t_{j}$ results in subtracting row $j$ to row $k$ , hence $j>k$ must hold. If we want to construct zeros in row $i=3$ (we cannot produce zeros in rows 1 and 2) in $S$ we need to find an (invertible) transformation matrix $Q\in\mathbb{K}^{n\times n}$ of the form (note that we will apply $Q^{-1}$ from the left)

[TABLE]

by solving the (underdetermined) linear system with the system matrix consisting of the rows $i+1,i+2,\ldots,n$ and the right hand side consisting of the row $i$ of $S$ :

[TABLE]

Here $\beta_{5}=-4/3$ and we choose $\beta_{4}=\beta_{6}=0$ . Now row 3 in $Q^{-1}S$ is zero

[TABLE]

that is $s^{\prime}_{3}=(Q^{-1}s)_{3}=0$ . Therefore we can remove row 3 and column 3 from the (admissibly) modified system $(uQ,AQ,v)$ and get one of dimension $n-1=5$ :

[TABLE]

The right family of the new ALS is $\bigl{(}1,x,0,1-\frac{4}{3}xy,x-\frac{4}{3}xyx\bigr{)}$ . After adding $4/3$ -times row 5 to row 3 we can remove row 3 and column 3 and get a minimal ALS for $3x-4xyx=(1-\frac{4}{3}xy)3x$ :

[TABLE]

4.2 Factorizing a Polynomial

Now we consider the following minimal ALS for $p=xyxyx+(3x-4xyx)$ , constructed in a similar way as shown in the previous subsection or using the right companion system (Definition 3.2):

[TABLE]

We try to create an upper right block of zeros of size $3\times 2$ . For that we apply the (admissible) transformation $(P,Q)$ directly to the coefficient matrices $A_{0}$ , $A_{x}$ and $A_{y}$ in $A=A_{0}\otimes 1+A_{x}\otimes x+A_{y}\otimes y$ to get the equations. For $y$ we have

[TABLE]

of which we pick the upper right $3\times 2$ block. Thus we have the following 6 equations for $y$ :

[TABLE]

Similarly, we get the following 6 equations for $x$ :

[TABLE]

And finally those for $1$ (without terms containing $\alpha_{3,4}=\beta_{4,5}=0$ ):

[TABLE]

A Gröbner basis for the ideal generated by these 18 equations (computed by FriCAS, [Fri16] using lexicographic order) is $\bigl{(}\alpha_{1,2}-\alpha_{1,4}\beta_{4,6},\;\alpha_{1,3}-\alpha_{1,5}\beta_{4,6},\;\alpha_{2,3}-\alpha_{2,5}\beta_{4,6},\;\alpha_{2,4}+3\beta_{4,6}-4,\;\alpha_{3,4},\;\alpha_{3,5}+3\beta_{4,6}-4,\;\beta_{2,5},\;\beta_{2,6}+1,\;\beta_{3,5}-3\beta_{4,6}+4,\;\beta_{3,6}-3\beta_{4,6}\beta_{5,6}+4\beta_{5,6},\;\beta_{4,5},\;\beta_{4,6}^{2}-\textstyle{\frac{4}{3}}\beta_{4,6}+\textstyle{\frac{1}{3}}\bigr{)}$ . The last (over $\mathbb{K}$ reducible) generator is $(\beta_{4,6}-1)(\beta_{4,6}-\frac{1}{3})$ , thus $\beta_{4,6}\in\{1,\frac{1}{3}\}$ . We proceed with the case $\beta_{4,6}=1$ . We then have $\alpha_{2,4}=\alpha_{3,5}=1$ , $\beta_{2,6}=\beta_{3,5}=-1$ and choose $\alpha_{1,2}=\alpha_{1,4}=0$ , $\alpha_{1,3}=\alpha_{1,5}=3$ , $\alpha_{2,3}=\alpha_{2,5}=0$ , $\beta_{3,6}=-\beta_{5,6}=0$ . Note that $\alpha_{4,5}$ , $\beta_{2,3}$ , $\beta_{2,4}$ and $\beta_{3,4}$ are not present, so we set them to zero. Therefore one possible transformation is

[TABLE]

leading to the admissible linear system $\mathcal{A}^{\prime}=P\mathcal{A}Q$

[TABLE]

compare with the ALS (4.1). $\mathcal{A}^{\prime}$ is the (minimal) product of

[TABLE]

and

[TABLE]

Thus $p=(xyx-x)(yx-3)$ . The first factor is not atomic because we could (pre-standard admissibly) construct either an $1\times 2$ upper right block of zeros (by subtracting $3$ -times row 3 from row 1) or an $2\times 1$ upper right block of zeros (by subtracting column 2 from column 4 and subtracting $2$ -times row 3 from row 1) in the system matrix of $\eqref{eqn:pf.ex.3}$ . On the other hand, a brief look at the ALS $\eqref{eqn:pf.ex.4}$ immediately shows that the second factor is irreducible.

Acknowledgement

I thank Franz Lehner for the fruitful discussions and his support, Daniel Smertnig who helped me to find some orientation in abstract non-commutative factorization (especially with literature), Roland Speicher for the creative environment in Saarbrücken and the anonymous referees for the valuable comments.

Bibliography31

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[ARJ 15] V. Arvind, G. Rattan, and P. Joglekar. On the complexity of noncommutative polynomial factorization. In Mathematical foundations of computer science 2015. Part II , volume 9235 of Lecture Notes in Comput. Sci. , pages 38–49. Springer, Heidelberg, 2015.
2[BHL 17] J. P. Bell, A. Heinle, and V. Levandovskyy. On noncommutative finite factorization domains. Trans. Amer. Math. Soc. , 369(4):2675–2695, 2017.
3[BK 00] L. A. Bokut ′ and P. S. Kolesnikov. Gröbner-Shirshov bases: from inception to the present time. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , 272(Vopr. Teor. Predst. Algebr i Grupp. 7):26–67, 345, 2000.
4[BR 11] J. Berstel and C. Reutenauer. Noncommutative rational series with applications , volume 137 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2011.
5[BS 15] N. R. Baeth and D. Smertnig. Factorization theory: from commutative to noncommutative settings. J. Algebra , 441:475–551, 2015.
6[Buc 70] B. Buchberger. Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems. Aequationes Math. , 4:374–383, 1970.
7[Car 10] F. Caruso. Factorization of non-commutative polynomials. Ar Xiv e-prints , February 2010.
8[CC 80] A. Cardon and M. Crochemore. Détermination de la représentation standard d’une série reconnaissable. RAIRO Inform. Théor. , 14(4):371–379, 1980.

TL;DR

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On the Factorization of

Abstract

Introduction

1 Representing Non-Commutative Polynomials

Definition 1.1** **(Inner Rank, Full Matrix, Hollow Matrix,

Definition 1.2** **(Associated Matrices,

Lemma 1.3** ([Coh95, Corollary 6.3.6]).**

Definition 1.4** (Linear Representations, Dimension, Rank [CR94, CR99]).**

Definition 1.5** **(Left and Right Families

Proposition 1.6** ([CR94], Proposition 4.7).**

Definition 1.7** (Admissible Linear Systems [Coh72]. Admissible Transformations).**

Proposition 1.8** (Rational Operations [CR99]).**

Definition 1.9**.**

Definition 1.10** **(Similar Right Ideals, Similar Elements

Definition 1.11** **(Left and Right Coprime Elements

Definition 1.12** **(Atomic Domains, Irreducible Elements

Definition 1.13** **(Similarity Unique Factorization Domains

Proposition 1.14** ([Coh63, Theorem 6.3]).**

2 Factorizing non-commutative Polynomials

Definition 2.1** (Pre-Standard ALS, Pre-Standard Admissible Transformation).**

2.1 Minimal Multiplication

Lemma 2.2**.**

Proof.

Lemma 2.3**.**

Proof.

Proposition 2.5** (Minimal Polynomial Multiplication).**

Proof.

Corollary 2.6**.**

Example 2.7**.**

2.2 Minimizing a pre-standard ALS

Definition 2.15** (Minimization Equations, Minimization Transformations).**

Algorithm 2.17** (Minimizing a pre-standard ALS).**

Proof.

Remark 2.18**.**

2.3 Factorizing non-commutative Polynomials

Definition 2.19** (Atomic Admissible Linear Systems).**

Definition 2.21** (Standard Admissible Linear Systems).**

Example 2.22**.**

Lemma 2.23**.**

Proof.

Theorem 2.24** (Polynomial Factorization).**

Proof.

Proposition 2.25**.**

3 Generalizing the Companion Matrix

Definition 3.1** **(Companion Matrix, Characteristic Polynomial, Normal Form

Definition 3.2** (Left and Right Companion System).**

Proposition 3.5**.**

Proof.

Definition 3.6** (Companion Matrices).**

Example 3.7**.**

4 An Example (step by step)

4.1 Constructing a minimal ALS

4.2 Factorizing a Polynomial

Acknowledgement

Definition 1.1 (Inner Rank, Full Matrix, Hollow Matrix,

Definition 1.2 (Associated Matrices,

Lemma 1.3 ([Coh95, Corollary 6.3.6]).

Definition 1.4 (Linear Representations, Dimension, Rank [CR94, CR99]).

Definition 1.5 (Left and Right Families

Proposition 1.6 ([CR94], Proposition 4.7).

Definition 1.7 (Admissible Linear Systems [Coh72]. Admissible Transformations).

Proposition 1.8 (Rational Operations [CR99]).

Definition 1.9.

Definition 1.10 (Similar Right Ideals, Similar Elements

Definition 1.11 (Left and Right Coprime Elements

Definition 1.12 (Atomic Domains, Irreducible Elements

Definition 1.13 (Similarity Unique Factorization Domains

Proposition 1.14 ([Coh63, Theorem 6.3]).

Definition 2.1 (Pre-Standard ALS, Pre-Standard Admissible Transformation).

Lemma 2.2.

Lemma 2.3.

Proposition 2.5 (Minimal Polynomial Multiplication).

Corollary 2.6.

Example 2.7.

Definition 2.15 (Minimization Equations, Minimization Transformations).

Algorithm 2.17 (Minimizing a pre-standard ALS).

Remark 2.18.

Definition 2.19 (Atomic Admissible Linear Systems).

Definition 2.21 (Standard Admissible Linear Systems).

Example 2.22.

Lemma 2.23.

Theorem 2.24 (Polynomial Factorization).

Proposition 2.25.

Definition 3.1 (Companion Matrix, Characteristic Polynomial, Normal Form

Definition 3.2 (Left and Right Companion System).

Proposition 3.5.

Definition 3.6 (Companion Matrices).

Example 3.7.