A Standard Form in (some) Free Fields: How to construct Minimal Linear Representations

We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in standard form). This completes"minimal"arithmetics in free fields since"minimal"constructions for the inverse are already known. Although in general it is difficult to transform a minimal linear representation into standard form, using rational operations"carefully"the form can be kept"close"to standard. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.


Introduction
While the embedding of the integers into the field of rational numbers is easy, that of non-commutative rings (into skew fields) is not that straight forward even in special cases [Ore31]. After Ore's construction it took almost forty years and many contributors to develop a more general theory [Coh06,Chapter 7]. The embedding of the free associative algebra (over a commutative field) into a ring of quotients (noncommutative localization) is even classified as "ugly" [Lam99].
On the other hand there are many parallels between the ring of integers and the free associative algebra, for example, both have a distributive factor lattice (DFL) [Coh82] or [Coh85,Section 3.5]. And, starting from the normal form (minimal linear representation) of Cohn and Reutenauer [CR94] of an element in the universal field of fractions (of the free associative algebra), we will formulate a standard form which can be seen as a "generalized" fraction. As a reminiscence to the work with "classical" fractions (we learn at school) we call them briefly "free fractions" [Sch18a].
For an introduction to free fields we recommend [Coh03,Section 9.3] and with respect to linear representations in particular [CR99]. For details we refer to [Coh06,Chapter 7] or [Coh95,Section 6.4]. Since this work is only one part in a series, further information and references can be found in [Sch18b] (linear word problem, minimal inverse), [Sch19] (polynomial factorization) and [Sch17] (general factorization theory).
The main idea of the standard form is simple: instead of viewing the system matrix of a linear representation as a single "block" we transform it into a block upper triangular form with smaller diagonal blocks. If these pivot blocks are small enough (called "refined"), linear techniques can be used to eliminate all superfluous block rows or columns, that is, solving "local" word problems, in the "sum" or the "product" of two linear representations, eventually yielding a minimal linear representation. This can be accomplished with complexity O(dn 5 ) for an alphabet with d letters and a linear representation of dimension n with pivot blocks of size √ n. For the refinement of pivot blocks we need to solve -at least in general-systems of polynomial equations. However linear techniques can be used in some cases to "break" big pivot blocks into smaller ones (see Remark 5.8).
Since almost all here is rather elementary it should be noted that it needs some effort to dig deep enough into the theory in the background (in particular that of Cohn) to really understand what is going on. Despite of the main results (mentioned in the following) there is only one non-trivial observation (formulated in Theorem 4.16): A given refined linear representation is minimal if non of the linear systems of equations for block row or column minimization has a solution. (In general nothing can be said about minimality if there is no more "linear" minimization step possible.) In other words: While the normal form [CR94] "linearizes" the word problem in free fields [Sch18b, Section 2], the standard form "linearizes" (the minimization of) the sum and the product (of two elements).
Section 1 provides the basic setup and Section 2 summarizes several (different) constructions of linear representations for the rational operations (sum, product and inverse). In a first reading only the first two propositions are important. In Section 3 we develop the notation to be able to formulate a standard form in Definition 3.8. The main result is then Theorem 4.16 (or Algorithm 4.14) for the minimization in Section 4. And finally, in Section 5, some applications are mentioned. Example 5.1 can also serve as an introduction for the work (by hand) with linear representations.
The intention of this paper is to be independent of the other three papers in this series (about the free field) as far as possible and leave it to the reader, for example, to interpret a standard form of the inverse of a polynomial as its factorization (into irreducible elements). Although the idea for minimizing "polynomial" linear representations is similar to that in Section 4, [Sch19, Algorithm 32] is only a very special case of Algorithm 4.14 and the "refinement" in the former case is trivial.
Beside the algebraic approach presented here there are analytical methods for solving the word problem (or testing rational identities like in Example 5.1) in polynomial time by plugging in "sufficiently large" matrices [GGOW16,IQS18]. Closely related to linear representations are realizations which can be "cut down" by plugging in operators [HMS18] or "reduced" by plugging in matrices [Vol18]. Yet another point of view is from invariant subspaces [GLR06,HKV18].
Once the rich structure of Cohn and Reutenauer's normal form becomes "visible" a lot can be done, transforming the rather abstract free field into an applied. Using non-commutative "rational functions" just like rational numbers respectively "free fractions" just like "classical" fractions . . .

Preliminaries
We represent elements (in free fields) by admissible linear systems (Definition 1.9), which are just a special form of linear representations (Definition 1.4). Rational operations (scalar multiplication, addition, multiplication, inverse) can be easily formulated in terms of linear representations (Proposition 2.2).
Notation. The set of the natural numbers is denoted by N = {1, 2, . . .}. Zero entries in matrices are usually replaced by (lower) dots to emphasize 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 and Σ n the permutation matrix that reverses the order of rows/columns (of size n) respectively I and Σ if the size is clear from the context. there exists a factorization A = T U with T ∈ K X n×m and U ∈ K X m×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 × l with k + l > n.
Definition 1.2 (Associated and Stably Associated Matrices [Coh95]). Two matrices A and B over K X (of the same size) are called associated over a subring R ⊆ K X if there exist invertible matrices P, Q over R such that A = P BQ. A and B (not necessarily of the same size) are called stably associated if A ⊕ I p and B ⊕ I q are associated for some unit matrices I p and I q . Here by C ⊕ D we denote the diagonal sum C .
. D . Lemma 1.3 ([Coh95, Corollary 6.3.6]). A linear square matrix over K X which is not full is associated over K to a linear hollow matrix.
It is called minimal if A has the smallest possible dimension among all linear representations of f . The "empty" representation π = (, , ) is the minimal one of 0 ∈ F with dim π = 0. Let f ∈ F and π be a minimal linear representation of f . Then the rank of f is defined as rank f = dim π.
Remark. Cohn and Reutenauer define linear representations slightly more general, namely f = c + uA −1 v with possibly non-zero c ∈ K and call it pure when c = 0. Two linear representations are called equivalent if they represent the same element [CR99]. Two (pure) linear representations (u, A, v) and (ũ,Ã,ṽ) of dimension n are called isomorphic if there exist invertible matrices P, Q ∈ K n×n such that u =ũQ, Theorem 1.5 ([CR99, Theorem 1.4]). If π ′ = (u ′ , A ′ , v ′ ) and π ′′ = (u ′′ , A ′′ , v ′′ ) are equivalent (pure) linear representations, of which the first is minimal, then the second is isomorphic to a representation π = (u, A, v) which has the block decomposition Remark. In principle, given π ′′ of dimension n, one can look for invertible matrices P, Q such that P π ′′ Q = (u ′′ Q, P A ′′ Q, P v ′′ ) has the form of π to minimize π ′′ . However, for an alphabet with d letters and a lower left block of zeros of size k × (n − k) we would get (d + 1)k(n − k) + k polynomial equations with at most quadratic terms and two equations of degree n (to ensure invertibility of the transformation matrices) in 2n 2 commuting unknowns. This is already rather challenging for n = 5. The goal is therefore to use linear techniques as far as possible.
Remark. The following definition is a special case of Cohn's more general admissible systems [Coh06, Section 7] and the slightly more general linear representations [CR94]. For elements in the free associative algebra K X a special form (with an upper unitriangular system matrix) can be used. It plays a crucial role in the factorization of polynomials because it allows to formulate a minimal polynomial multiplication (Proposition 2.12) and upper unitriangular transformation matrices (invertible by definition) suffice to find all possible factors (up to trivial units). For details we refer to [Sch19, Section 2].
Remark 1.10. While it was intended in [Sch19] to derive a "better" standard form including knowledge about the factorization it turned out later that this is not that easy in the general case because of the necessity to distinguish several cases in the multiplication [Sch17, Section 4]. The term "pre-standard ALS" (for polynomials) is now replaced by polynomial ALS (which is just a special form of a refined ALS, Section 3). And a minimal polynomial ALS is already in standard form. Especially to avoid confusion with special transformation matrices for the factorization everything is put into a uniform context in [Sch18c]. For an overview see also [Sch18a, Figure 1]. There are only some minor changes in the formulation of results and proofs necessary, for example to construct the product of q 1 q 2 in [Sch19, Lemma 39] using λ2 λ A 1 and λ λ2 A 2 . Definition 1.11 (Polynomial ALS and Transformation [Sch19]). An ALS A = (u, A, v) of dimension n with system matrix A = (a ij ) for a non-zero polynomial 0 = p ∈ K X is called polynomial, if (2) a ii = 1 for i = 1, 2, . . . , n and a ij = 0 for i > j, that is, A is upper triangular.

Rational Operations
Usually we want to construct minimal admissible linear systems (out of minimal ones), that is, perform "minimal" rational operations. Minimal scalar multiplication is trivial. In some special cases minimal multiplication or even minimal addition (if two elements are disjoint [CR99] Proposition 2.2 (Rational Operations [CR99]). Let 0 = f, g ∈ F be given by the admissible linear systems A f = (u f , A f , v f ) and A g = (u g , A g , v g ) respectively and let 0 = µ ∈ K. Then admissible linear systems for the rational operations can be obtained as follows: The scalar multiplication µf is given by The sum f + g is given by The product f g is given by And the inverse f −1 is given by 1 .
Since we need alternative constructions (to that in Proposition 2.2) for the product we state them here in Propositions 2.6 and 2.7. Before, we need some technical results from [Sch18b] and [Sch19]. However these are rearranged such that similarities become more obvious and the flexibility in applications is increased. In particular one can proof Lemma 2.4 by applying Lemma 2.3, see [Sch17, Lemma 2.6].
Remark. If g is of type ( * , 1) then, by Lemma 2.4, each minimal ALS for g can be transformed into one with a last row of the form [0, . . . , 0, 1]. If g is of type (1, * ) then, by Lemma 2.5, each minimal ALS for g can be transformed into one with a first column of the form [1, 0, . . . , 0] ⊤ . This can be done by linear techniques, see the remark before [Sch18b, Theorem 4.13].
Remark. Since p ∈ K X is of type (1, 1), both constructions can be used for the minimal polynomial multiplication (Proposition 2.12). The alternative proof in [Sch17] relies in particular on Lemma 2.11. One could call the multiplication from Proposition 2.2 type ( * , * ). For a discussion of minimality of different types of multiplication we refer to [Sch17].
Then an ALS for f g of dimension n = n f + n g − 1 is given by Proposition 2.7 (Multiplication Type ( * , 1) [Sch17, Proposition 2.11]). Let f, g ∈ F \ K be given by the admissible linear systems respectively. Then an ALS for f g of dimension n = n f + n g − 1 is given by Remark. Notice that the transformation in the following lemma is not necessarily admissible. However, except for n = 2 (which can be treated by permuting the last two elements in the left family), it can be chosen such that it is admissible. The proof is similar to that of [Sch19, Lemma 2.4].
Lemma 2.10 ([Sch17, Lemma 2.14]). Let A = (u, A, v) be an ALS of dimension n ≥ 2 with v = [0, . . . , 0, λ] ⊤ and K-linearly dependent left family s = A −1 v. Let m ∈ {2, 3, . . . , n} be the minimal index such that the left subfamily s = (A −1 v) n i=m is K-linearly independent. Let A = (a ij ) and assume that a ii = 1 for 1 ≤ i ≤ m and a ij = 0 for j < i ≤ m (upper triangular m×m block) and a ij = 0 for j ≤ m < i (lower left zero block of size (n − m) × m). Then there exists matrices T, Lemma 2.11 ([Sch17, Lemma 2.15]). Let p ∈ K X \ K and g ∈ F \ K be given by the minimal admissible linear systems A p = (u p , A p , v p ) and A g = (u g , A g , v g ) of dimension n p and n g respectively with 1 ∈ R(g). Then the left family of the ALS A = (u, A, v) for pg of dimension n = n p + n g − 1 from Proposition 2.7 is K-linearly independent.
Proposition 2.12 (Minimal Polynomial Multiplication [Sch19, Proposition 26]). Let 0 = p, q ∈ K X be given by the minimal polynomial admissible linear systems A p = (1, A p , λ p ) and A q = (1, A q , λ q ) of dimension n p , n q ≥ 2 respectively. Then the ALS A from Proposition 2.6 for pq is minimal of dimension n = n p + n q − 1.
Theorem 2.13 (Minimal Inverse [Sch18b, Theorem 4.13]). Let f ∈ F \ K be given by the minimal admissible linear system A = (u, A, v) of dimension n. Then a minimal ALS for f −1 is given in the following way: (2.14) f of type (1, 0) yields f −1 of type (1, 0) with dim(A ′ ) = n: (Recall that the permutation matrix Σ reverses the order of rows/columns.) Corollary 2.18. Let p ∈ K X with rank p = n ≥ 2. Then rank(p −1 ) = n − 1.

A Standard Form
After providing a "language" to be able to formulate "operations" on the system matrix of an admissible linear system we can define a standard form (Definition 3.8).
A standard ALS will be minimal and (has) refined (pivot blocks). In the following section we will minimize a refined ALS. This is somewhat technical to describe but simple (we need to solve linear systems of equations). At the end of this section we will illustrate how to refine pivot blocks. To the contrary this is easy to describe but in general very difficult to accomplish (we need to solve polynomial systems of equations). Since the latter is closely related to factorization we refer to [Sch19] and [Sch17] for further information.
Notation. Let A = (u, A, v) be an ALS with m pivot blocks of size n i = dim i (A). We denote by n i:j = dim i:j (A) = n i + n i+1 + . . . + n j the sum of the sizes of the pivot blocks A ii to A jj (with the convention n i:j = 0 for j < i). For a given system the identity matrix of size n i:j is denoted by I i:j . If (P, Q) is an admissible transformation for A then (P AQ) ij denotes the (to the block decomposition of A corresponding) block (i, j) of size n i × n j in P AQ, (P v) i that of size n i × 1 in P v and (uQ) j that of size 1 × n j in uQ.
Notation. Components in the left family . A subfamily of s with respect to the pivot block k is denoted by s k , s i:j = (s i , s i+1 , . . . , s j ). Analogous is used for the right family t = uA −1 .

Notation.
A "grouping" of pivot blocks {i, i + 1, . . . , j} of the system matrix is denoted by A i:j,i:j . If it is clear from the context where the block ends (respectively starts) we write A i:,i: (respectively A :j,:j ), in particular with respect to a given pivot block. For example A 1:,1: , A k,k and A :m,:m .

Definition 3.3 (Refined Pivot Block and Refined ALS). Let
is called refined if there does not exist an admissible pivot block transformation (P, Q) k such that (P AQ) kk has a lower left block of zeros of size i × (n k − i) for an i ∈ {1, 2, . . . , n k − 1}. The admissible linear system A is called refined if all pivot blocks are refined.
Remarks. That the "form" of a refined ALS is not unique will be illustrated in the following example. And a refined ALS is not necessarily refined over K: Let Adding √ 2-times row 1 to row 2 and subtracting √ 2-times column 2 of column 1 yields 1 .
For f + 3z we have the (minimal) ALS Since 1 ∈ R(A) and A is constructed by the addition in Proposition 2.2, it can easily be transformed -in a controlled way-into another ALS with refined pivot block structure. Firstly we add column 3 to column 1, and then we exchange rows 1 and 3: is called k-th block column transformation for A [k] . ForT =Ū = I n k we write also P (T ) respectively Q(U ) and call the block transformation particular. Remark. For a polynomial p given by a standard ALS A (of dimension n ≥ 2) the minimal inverse of A (of dimension n − 1) is refined if and only if A is obtained by the minimal polynomial multiplication of its irreducible factors q i in p = q 1 q 2 · · · q m .
For a detailed discussion about the factorization of polynomials (in free associative algebras) we refer to [Sch19]. One of the simplest non-trivial examples is With respect of the general factorization theory it is open to show that the free field is a "similarity unique factorization domain" [Sch17].
Later, in Example 5.1 (Hua's identity), we will need to refine a pivot block. Although the necessary transformations there are obvious the procedure should be illustrated in a systematic way. But before we have a look on how this 2 × 2 block appeared, namely by inverting the element given by the ALS If we exchange columns 2 and 3 it is immediate that this is the "product" of the admissible linear systems Applying the minimal inverse on we get a refined (and minimal) ALS, namely The factorization here is really simple. Now we focus on the refinement of the second pivot block in the ALS (This is the ALS (5.3) with the first three rows scaled by −1.) We are looking for an admissible transformation (P, Q) of the form In particular these matrices P and Q have to be invertible, that is, we need the conditions det(P ) = 0 and det(Q) = 0. To create a lower left 1 × 1 block of zeros in (P AQ) 2,2 we need to solve the following polynomial system of equations (with commuting unknowns α ij and β ij ): α 2,2 α 3,3 − α 2,3 α 3,2 = 1, β 2,2 β 3,3 − β 2,3 β 3,2 = 1, α 3,2 β 3,2 + α 3,3 β 2,2 = 0 for 1, and α 3,2 β 2,2 = 0 for y.
We obtain the last two equations by multiplication of the transformation blocks with the corresponding coefficient matrices of the pivot blocks (irrelevant equations are marked with " * " on the right hand side) α 2,2 α 2,3 α 3,2 α 3,3 . 1 1 .
To solve this system of polynomial equations, Gröbner-Shirshov bases [BK00] can be used. For detailed discussions we refer to [Stu02] or [CLO15]. .
Now we can read off a sufficient condition for (Q −1 s) k = 0 n k ×1 , namely the existence of matrices T, U ∈ K n k ×n k+1:m and invertible matricesT ,Ū ∈ K n k ×n k such that T A k,k U +T A k,:m + T A :m,:m = 0 n k ×n k+1:m andT v k + T v :m = 0 n k ×1 .
SinceT is invertible (as a diagonal block of an invertible matrix P ), this condition is equivalent to the existence of matrices T ′ , U ∈ K n k ×n k+1:m such that Remark 4.3 (Extended ALS). In some cases it is necessary to use an extended ALS to be able to apply all necessary left minimization steps, for example, for f −1 f if f is of type (1, 1). Let A = (u, A, v) = (1, A, λ) be an ALS with m = # pb (A) ≥ 2 pivot blocks and k = 1. The "extended" block decomposition is then (the block row A 1:,1: vanishes) .
whose pivot blocks are refined. Here there exists an admissible transformation (with T = 0, U = 1 and invertible blocksT ,Ū ∈ K 3×3 ) that yields the ALS in which one can eliminate row 3 and column 3 (and -after an appropriate row operation-also the last row and column). However, minimization can be accomplished much easier: Firstly we observe that the left subfamily s 2:3 (of A) is K-linearly independent. Also the right subfamily t 1:2 is K-linearly independent. For the left family with respect to the first pivot block we consider the extended ALS (see also Remark 4.3) of A, the upper row and the left column is indexed by zero. Now we add row 3 to row 1, subtract column 1 from column 3 and add column 1 to column 4 (this transformation can be found by solving a linear system of equations): Now we can remove row 1 and column 1: Before we do the last (right) minimization step, we transform the extended ALS back into a "normal" by exchanging columns 1 and 2, scaling the (new) column 1 by −1 and subtract it from column 3:  (If necessary right minimization steps can be performed until one reaches a pivot block with corresponding non-zero entry in row 0.) Now we can remove row 0 and column 0 again. The last step to a minimal ALS for f g = (1 − yx) −1 is trivial: After removing the last row and column we exchange the two rows to get a standard ALS.
Now at least one question should have appeared: How can one prove -for a given block index k-the K-linear independence of the left (respectively right) subfamily s k:m (respectively t 1:k ) in general, assuming that s k+1:m (respectively t 1:k−1 ) is Klinearly independent? For an answer some preparation is necessary. Lemma 1.2]). Let f ∈ F given by the linear representation π f = (u, A, v) of dimension n. Then f = 0 if and only if there exist invertible matrices P, Q ∈ K n×n such that for square matricesÃ 1,1 andÃ 2,2 .
If the pivot block A k,k is refined, then there exists a particular block row transformation (P, Q) = P (T ), Q(U ) k , such that the left block minimization equations Due to the K-linear independence of the left subfamily s k+1:m there exists an invertible matrixQ with blocksŪ • ∈ K n k ×n k and U • ∈ K n k ×n k+1:m , such that (Q −1 s) n 1:k−1 +1 = 0, that is, the first component in s k can be eliminated. Let and v ′ = . v :m .  For the second part we first have to show that each component in s k can be eliminated by a linear combination of components of s k+1:m , that is, n k ′′ = 0. We assume to the contrary that n k ′′ > 0. But then -by (4.9)-T A k,kŪ would have an upper right block of zeros of size (n k −n k ′′ )×n k ′′ and therefore (after an appropriate permutation) a lower left, contradicting the assumption on a refined pivot block. Hence there exists a matrix U ∈ K n k ×n k+1:m such that s k − U [s k+1 , . . . , s m ] = 0. By assumption v k = 0. Now we can apply -as in Lemma 2.10-Lemma 2.3 with the ALS (1, A :m,:m , λ) and B = −A k,k U − A k,:m (and s :m ). Thus there exists a matrix T ∈ K n k ×n k+1:m fulfilling A k,k U + A k,:m + T A :m,:m = 0. Since the last column of T is zero, we have also T v :m = 0. WithT =Ū = I 1:n k the transformation (P, Q) is the appropriate particular block row transformation.
Remark. For the proof of the second part of the theorem one can use alternatively Lemma 4.7 which is more powerful but with respect to the use of linear techniques not that obvious.
Remark. Notice that the left subfamily (s k ′′ , s :m ) is not necessarily K-linearly independent. If necessary, one can apply the theorem again after removing block row and column k ′ .
Remark. For k = 1, if necessary, one must use an extended ALS, see Remark 4.3. . , m} such that the right subfamily t 1:k−1 with respect to the block decomposition A [k] is K-linearly independent while t 1:k is K-linearly dependent. Then there exists a block column transformation (P, Q) = P (T , T ), Q(Ū, U ) k , such that A = P AQ has the form If the pivot block A k,k is refined, then there exists a particular block column transformation (P, Q) = P (T ), Q(U ) k , such that the right block minimization equations A 1:,1: U + A 1:,k + T A k,k = 0 n 1:k−1 ×n k are fulfilled.
If one uses alternating left and right block minimization steps for the minimization, that is, applying Theorems 4.8 and 4.11, one has to take care that the K-linear independence of the respective other subfamily is guaranteed. This is illustrated in the following example.
Example 4.13. Let A = (u, A, v) = (1, A, λ) be an ALS with m = 5 pivot blocks. For k ′ = 2 we assume that the left subfamily s k+1:m is K-linearly independent and we assume further that there exists a particular block row transformation (P, Q), such that the left block minimization equations are fulfilled, that is, P AQ has the form If the right subfamily t 1:3 is K-linearly independent, this is not necessarily the case for the right subfamily t ′ 1:3 of the smaller ALS A ′ = P AQ [−k ′ ] . That is, one has to apply Theorem 4.11 on A ′ with k = 3 to check that "again". return P A, with P , such that P v = [0, . . . , 0, λ ′ ] ⊤ Proof. The admissible linear system A represents f = 0 if and only if s 1 = (A −1 v) 1 = 0. Since all systems are equivalent to A, this case is recognized for k ′ = 1 because by Theorem 4.8 there is an admissible transformation such that the first left block minimization equation is fulfilled. Now assume f = 0. We have to show that both, the left family s ′ and the right family t ′ of A ′ = (u ′ , A ′ , v ′ ) are K-linearly independent respectively. Let m ′ = # pb (A ′ ) and for k ∈ {1, 2, . . . , m ′ } denote by the left and the right subfamily respectively. By assumption s ′ (1) and t ′ (1) are Klinearly independent respectively. The loop starts with k = 2. Only if both s ′ (k) and t ′ (k) are K-linearly independent respectively, k is incremented. Otherwise a left (Theorem 4.8) or a right (Theorem 4.11) 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. (How row 0 and column 0 are removed from the extended ALS in line 14 is illustrated in Example 4.5.) All transformations are such that A ′ is a refined ALS (and therefore in standard form). For # pb (A ′ ) = 1 a priori only the left (or the right) family is Klinearly independent (by assumption). But if that were not the case for the respective other family, then the assumption on refined pivot blocks would be contradicted by Theorem 4.11 (respectively Theorem 4.8) Remark. Concerning details with respect to the complexity of such an algorithm we refer to [Sch19,Remark 33]. Let d be the number of letters in our alphabet X. For m = n pivot blocks and k < n we have 2(k − 1) unknowns. By Gaussian elimination one gets complexity O(dn 3 ) for solving a linear system for a minimization step, see [Dem97,Section 2.3]. To build such a system and working on a linear matrix pencil 0 u v A with d + 1 square coefficient matrices of size n + 1 (transformations, etc.) has complexity O(dn 2 ). So we get overall (minimization) complexity O(dn 4 ). The algorithm of Cardon and Crochemore [CC80] has complexity O(dn 3 ) but works only for regular elements, that is, rational formal power series. For dim i (A) ≈ √ n we get complexity O(dn 5 ) and for the word problem [Sch18b] with m = 2 we get complexity O(dn 6 ).
Remark. It is clear that one can adapt the algorithm slightly if the input ALS is constructed by Proposition 2.2 out of two minimal admissible linear systems (for the sum and the product) in standard form.
Remark. The solution of the word problem for two elements given by minimal admissible linear systems is independent of their refinement. If Algorithm 4.14 is applied to an ALS of which it is not known if it is refined, in some cases it is possible to check if the ALS A ′ is minimal, for example if dim(A ′ ) = # pb (A ′ ). If the pivot blocks are bigger but the right upper structure is "finer" one can instead -for f = 0-try to minimize the inverse (A ′ ) −1 . In concrete situations there might be other possibilities to reach minimality. Another aspect of Algorithm 4.14 (respectively Theorem 4.8 and 4.11) becomes visible immediately with Proposition 1.7 and Remark 4.10. The importance of the following theorem becomes clear if one needs to check K-linear (in-)dependence of an arbitrary family (f 1 , f 2 , . . . , f n ) over the free field and it is not possible (anymore) to take a representation as formal power series. Proof. From the existence of a solution non-minimality follows immediately since in this case -after the appropriate transformation-rows and columns can be removed. And for non-minimality Proposition 1.7 implies that either the left or the right family is K-linearly dependent. Without loss of generality assume that it is the left s = (s 1 , s 2 , . . . , s m ) with minimal k ∈ {1, 2, . . . , m − 1} such that the left subfamily (s k+1 , . . . , s m ) is K-linearly independent. Since the pivot blocks are refined, Theorem 4.8 implies the existence of a particular block row transformation (P, Q) k and therefore a solution of the k-th left block minimization equations.

Applications
Since the focus of this work is mainly minimization and one dedicated to "minimal" rational operations -collecting all techniques for practical application-is already available [Sch18a], only two applications are illustrated in the following examples. For other applications see also Remark 4.15.
Example 5.1 (Hua's Identity [Ami66]). We have: Proof. Minimal admissible linear systems for y −1 and x are y s = 1 and respectively. The ALS for the difference y −1 − x, is minimal because the left family s is K-linearly independent and the right family t is K-linearly independent (Proposition 1.7). Clearly we have 1 ∈ R(y −1 − x). Thus, by Lemma 2.5, there exists an admissible transformation Now we can apply the inverse of type (1, 1): This system represents a regular element (y −1 − x) −1 = (1 − yx) −1 y, and therefore can be transformed into a regular ALS (Definition 1.13) by scaling row 2 by −1. Then we add x −1 "from the left": This system is minimal and -after adding row 3 to row 1 (to eliminate the non-zero entry in the right hand side)-we apply the (minimal) inverse of type (0, 0):  Now we multiply row 1 and the columns 2 and 3 by −1 and exchange column 2 and 3 to get the following system:  The next step would be a scaling by −1 and the addition of x (by Proposition 2.2).
With two minimization steps we would reach again minimality. Alternatively we can add a linear term to a polynomial (in a polynomial ALS) -depending on the entry v n in the right hand side-directly in the upper right entry of the system matrix: Remarks. The transformation of the ALS (5.3) is a simple case of the refinement of a pivot block and is discussed in detail in Section 3. Hua's identity is also an example in [CR94]. It is worth to compare both approaches.
Example 5.4 (Left GCD). Given two polynomials p, q ∈ K X \ K, one can compute the left (respectively right ) greatest common divisor of p and q by minimizing an admissible linear system for p −1 q (respectively pq −1 ). This is now illustrated in the following example. Let p = yx(1 − yx)z = yxz − yxyxz and q = y(1 − xy)y = y 2 − yxy 2 . We want to find h = lgcd(p, q). An ALS for p −1 q (constructed out of minimal admissible linear systems for p −1 and q by Proposition 2.6) is Clearly, this system is refined. How to refine an ALS is discussed in Section 3. Note, that there is a close connection to the factorization of p, for details see [Sch19]. As a first step we add column 5 to column 6 and row 6 to row 4, , remove rows and columns 5 and 6, and remember the first (left) divisor h 1 = y we have eliminated. (In the next step with a bigger block one can see immediately how to "read" a divisor directly from the ALS.) Now there are two ways to proceed: If it is not possible to create a zero block in "L"-form (like before), one can try to change the upper pivot block structure to create a "double-L" zero block. Here, this is possible by subtracting column 4 from column 2 and adding row 2 to row 4. (For details on similarity unique factorization in this context see [Sch19].) Afterwards we apply the (admissible) transformation How to get this transformation (P, Q) is described in principle in Section 4. One can look directly for a "double-L" block transformation. In the third pivot block of (5.6) one can see immediately that a further (common) left factor is h 2 = 1 − xy because the second equation reads xs 2 − h −1 2 = 0. We have eliminated (1 − xy) −1 (1 − xy). Recall that a minimal ALS for h 2 is hence rank(h 2 ) = 3 and (by Theorem 2.13) rank(h −1 2 ) = 2. Or, more general, for a (left) factor h i with rank(h i ) = n i ≥ 2 we can construct (by Proposition 2.6) an ALS of dimension 2(n i − 1). After removing rows and columns {3, 4, 5, 6} in the ALS (5.6), we obtain for p −1 q the (in this case minimal ) ALS (5.7) Hence h = h 1 h 2 = y(1 − xy) = lgcd(p, q). The second possibility -starting from ALS (5.5)-is to do a right minimization step with respect to column 5, then one left with respect to rows 2 and 3 and finally a right (minimization step). Again one obtains the ALS (5.7) (up to admissible scaling of rows and columns) where the right factor y of q remains. Therefore lgcd(p, q) = y − yxy. For further details concerning the minimal polynomial multiplication (Proposition 2.12) we refer to [Sch19].
Remark. It can happen that -after there is no more "L"-minimization step possible-the ALS is not minimal, that is, an additional "single" left or right minimization step can be carried out. More details on that are part of the general factorization theory [Sch17]. Here it suffices to take a closer look at the ALS (5.7): both right factors of p (here xz) respectively q (here y) can still be "read" directly. (This would not be possible any more if one left or right step would be carried out.) Remark 5.8. For a refined ALS for p −1 , an ALS of dimension n for p −1 q with "factors" of rank √ n and an alphabet with d letters, the complexity for computing the left (or right) gcd is roughly O(dn 5 ). Although in general refinement is difficult because of the necessity to solve systems of polynomial equations (over a not necessarily algebraically closed field), especially for polynomials linear techniques are very useful for the factorization (and therefore for the refinement of the inverse). As an example we take the polynomial p = (1 − xy)(2 + yx)(3 − yz)(2 − zy)(1 − xz)(3 + zx)x of rank n = 14 which has already 64 terms. To get the first left (irreducible) factor (of rank 3) we just need to create an upper right block of zeros (in the system matrix) of size 2 × 11 which can be accomplished by either using the columns 2-3 and rows 4-13 or column 2 and rows 3-13 (for elimination) [Sch19]. Both cases result in a linear system of equations because column and row transformations do not "overlapp". Solving one of these 2(n − 2) systems has (at most) complexity O(dn 6 ), hence in total we have O(dn 7 ). Checking irreducibility of polynomials (using Gröbner bases) works practically up to rank 12 [Jan18].

Epilogue
This work is the last in a series for the development of tools for the work with linear representations (for elements in the free field) especially for the implementation in computer algebra systems. A "practical" guide giving an overview and an introduction is [Sch18c] (in German, with remarks on the implementation) and [Sch18a].
At some point one stops this loop and uses the fraction (with coprime numerator and denominator). Clearly, one could simplify things by remembering the factorization of the numerator (for the product) and the denominator (for the sum and the product).
In our case of the free field, linear representations (or admissible linear systems) are just "free fractions" . . . However, to understand how the transition from using nc rational expressions (to represent elements in the free field) to (minimal) admissible linear systems in standard form effects the capabilities of thinking (free) nc algebra, one needs to go to the meta level [Krä14].