Computing the Hermite Form of a Matrix of Ore Polynomials
Abstract
Let be the ring of Ore polynomials over a field (or a skew field) , where is an automorphism of and is a derivation. Given a matrix , we show how to compute the Hermite form of and a unimodular matrix such that . The algorithm requires a polynomial number of operations in in terms of the dimensions and , and the degrees (in ) of the entries in . When for some field , it also requires time polynomial in the degrees in of the coefficients of the entries, and if it requires time polynomial in the bit length of the rational coefficients as well. Explicit analyses are provided for the complexity, in particular for the important cases of differential and shift polynomials over . To accomplish our algorithm, we apply the Dieudonné determinant and quasideterminant theory for Ore polynomial rings to get explicit bounds on the degrees and sizes of entries in and .
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1 Introduction
The Ore polynomials are a natural algebraic structure which captures difference, difference, differential, and other noncommutative polynomial rings. The basic concepts of pseudolinear algebra are presented nicely by Bronstein and Petkovšek (1996); see (Ore, 1931) for the seminal introduction.
On the other hand, canonical forms of matrices over commutative principal ideal domains (such as or , for a field ) have proven invaluable for both mathematical and computational purposes. One of the successes of computer algebra over the past three decades has been the development of fast algorithms for computing these canonical forms. These include triangular forms such as the Hermite form (Hermite, 1851), low degree forms like the Popov form (Popov, 1972), as well as the diagonal Smith form (Smith, 1861).
Canonical forms of matrices over noncommutative domains, especially rings of differential and difference operators, are also extremely useful. These have been examined at least since the work of Dickson (1923), Wedderburn (1932), and Jacobson (1943). Recently they have found uses in control theory (Chyzak et al., 2005; Zerz, 2006; Halás, 2008). Computations with multidimensional linear systems over Ore algebras are nicely developed by Chyzak et al. (2007), and an excellent implementation of many fundamental algorithms is provided in the OreModules package of Maple.
In this paper we consider canonical forms of matrices of Ore polynomials over a skew field . Let be an automorphism of and be a derivation. That is, for any , and . We then define as the set of usual polynomials in under the usual addition, but with multiplication defined by
for any . This is wellknown to be a left (and right) principal ideal domain, with a straightforward Euclidean algorithm (see (Ore, 1933)).
Some important cases over the field of rational functions over a field are as follows:

is a socalled shift automorphism of , and identically zero on . Then is generally referred to as the ring of shift polynomials. With a slight abuse of notation we write for this ring.

and , so for any with its usual derivative. Then is called the ring of differential polynomials. With a slight abuse of notation we write for this ring.
A primary motivation in the definition of is that there is a natural action on the space of infinitely differentiable functions in , namely the differential polynomial
acts as the linear differential operator
on an infinitely differentiable function . See (Bronstein and Petkovšek, 1996).
The (row) Hermite form we will compute here is achieved purely by row operations, and we treat a matrix as generating the left module of its rows. Thus, by left row rank, we mean the rank of the free left module of rows of , and will denote this simply as the rank of for the remainder of the paper. A matrix of rank is in Hermite form if an only if

Only the first rows are nonzero;

In each row the leading (first nonzero) element is monic;

All entries in the column below the leading element in any row are zero;

All entries in the column above the leading element in any row are of lower degree than the leading element.
For square matrices of full rank the Hermite form will thus be upper triangular with monic entries on the diagonal, whose degrees dominate all other entries in their column.
For example, in the differential polynomial ring as above:
(1.1) 
has Hermite form
Note that the Hermite form may have denominators in . Also, while this example does not demonstrate it, the degrees in the Hermite form, in both numerators and denominators in and , are generally substantially larger than in the input (in Theorem 25 we will provide polynomial, though quite large, bounds on these degrees, and suspect these bounds may well be met generically).
For any matrix of full rank, there exists a unique unimodular matrix (i.e., a matrix whose inverse exists and is also in ) such that is in Hermite form. This form is canonical in the sense that if two matrices are such that for unimodular then the Hermite form of equals the Hermite form of . Existence and uniqueness of the Hermite form are established much as they are over in Section 2. For rank deficient matrices and rectangular matrices, the Hermite form also exists but the transformation matrix may not be unique. See Section 6 for further details.
In commutative domains such as and there have been enormous advances in the past two decades in computing Hermite, Smith and Popov forms. Polynomialtime algorithms for the Smith and Hermite forms over were developed by Kannan (1985), with important advances by Kaltofen et al. (1987), Villard (1995), Mulders and Storjohann (2003), Pernet and Stein (2010), and many others. One of the key features of this recent work in computing canonical forms has been a careful analysis of the complexity in terms of matrix size, entry degree, and coefficient swell. Clearly identifying and analyzing the cost in terms of all these parameters has led to a dramatic drop in both theoretical and practical complexity.
Computing the classical Smith and Hermite forms of matrices over Ore domains has received less attention though canonical forms of differential polynomial matrices have applications in solving differential systems and control theory (see (Halás, 2008; Kotta et al., 2008)). Abramov and Bronstein (2001) analyze the number of reduction steps necessary to compute a rowreduced form, while Beckermann et al. (2006) analyze the complexity of row reduction in terms of matrix size, degree and the sizes of the coefficients of some shifts of the input matrix. Beckermann et al. (2006) demonstrate tight bounds on the degree and coefficient sizes of the output, which we will employ here. For the Popov form, Cheng (2003) gives an algorithm for matrices of shift polynomials. Cheng’s approach involves order bases computation in order to eliminate lower order terms of Ore polynomial matrices. A main contribution of Cheng (2003) is to give an algorithm computing the rank and a rowreduced basis of the left nullspace of a matrix of Ore polynomials in a fractionfree way. This idea is extended in Davies et al. (2008) to compute the Popov form of general Ore polynomial matrices. They reduce the problem of computing Popov form to a nullspace computation. However, though Popov form is useful for rewriting high order terms with respect to low order terms, we want a different canonical form more suited to solving system of linear diophantine equations. Since the Hermite form is upper triangular, it meets this goal nicely, not to mention the fact that it is a “classical” canonical form. An implementation of the basic (exponentialtime) Hermite algorithm is provided by Culianez (2005). In (Giesbrecht and Kim, 2009), we present a polynomialtime algorithm for the Hermite form over , for full rank square matrices. While it relies on similar techniques as this current paper, the cost of the algorithm is higher, the coefficient bounds weaker, and it does not work for matrices of general Ore polynomials.
The related “twosided” problem of computing the Jacobson (noncommutative Smith) canonical form has also been recently considered. Blinkov et al. (2003) implement the standard algorithm in the package Janet. Levandovskyy and Schindelar (2011) provide a very complete implementation, for the full Ore case over skew fields, of a Jacobson form algorithm using Gröbner bases in Singular. Middeke (2008) has recently demonstrated that the Jacobson form of a matrix of differential polynomials can be computed in time polynomial in the matrix size and degree (but the coefficient size is not analyzed). Giesbrecht and Heinle (2012) give a probabilistic polynomialtime algorithm for this problem in the differential case.
One of the primary difficulties in both developing efficient algorithms for matrices of Ore polynomials, and in their analysis, is the lack of a standard notion of determinant, and the important bounds this provides on degrees in eliminations. In Section 3 we establish bounds on the degrees of entries in the inverse of a matrix over any noncommutative field with a reasonable degree function. We do this by introducing the quasideterminant of Gel’fand and Retakh (1991, 1992) and analyzing its interaction with the degree function. We also prove similar bounds on the degree of the Dieudonné determinant. In both cases, the bounds are essentially the same as for matrices over a commutative function field.
In Section 4 we consider matrices over the Ore polynomials and bound the degrees of entries in the Hermite form and corresponding unimodular transformation matrices. We also bound the degrees of the Dieudonné determinants of these matrices.
In Section 5 we present our algorithm for the Hermite form. The degree bounds and costs of our algorithms are summarized as follows, from Theorems 17, 19 and 25.
Summary Theorem.
Let have full rank with entries of degree at most in , and coefficients of degree at most in . Let be the Hermite form of and such that .

The sum of degrees in in any row of is at most , and each entry in has degree in at most .

All coefficients from of entries of and have degrees in , of both numerators and denominators, bounded by .

We can compute and deterministically with ^{2}^{2}2We employ softOh notation: for functions and we say if for some constant ., operations in .

Assume has at least elements. We can compute the Hermite form and with an expected number of of operations in using standard polynomial arithmetic. This algorithm is probabilistic of the Las Vegas type; it never returns an incorrect answer.
The cost of our algorithm for Ore polynomials over an arbitrary skew field, as well as over more specific fields like , is also shown to be polynomially bounded, and is discussed in Section 5.
It should be noted from the above theorem that the output is of quite substantial size. The transformation matrix as above is an matrix of polynomials in of degree bounded by in and each coefficient has degree bounded by , for a total size of elements of . While we have not proven our size bounds are tight, we have some confidence they are quite strong.
The algorithm presented in Section 5 is derived from the “linear systems” approach of Kaltofen et al. (1987) and Storjohann (1994). In particular, it reduces the problem to that of linear system solving over the generally commutative ground field (e.g., ). There are efficient algorithms and implementations for solving this problem. While we expect that further algorithmic refinements and reductions in cost can be achieved before an industrialstrength implementation is made, the general approach of reducing to wellstudied computational problem in a commutative domain would seem to have considerable merit in theory and practice.
In Section 6 we show that for the case of rankdeficient and rectangular matrices, the computation of the Hermite form is reduced to the full rank, square case.
2 Existence and Uniqueness of the Hermite form over Ore domains
In this section we establish the basic existence and uniqueness of Hermite forms over Ore domains. These follow similarly to the traditional proofs over ; see for example (Newman, 1972, Theorems II.2 and II.3), which we outline below.
Fact 1 (Jacobson (1943), Section 3.7).
Let , not both zero with , such that , and such that . Then
and is unimodular.
This is easily generalized to matrices as follows.
Lemma 2.
Let , and . There exists a matrix
such that , with and .
Proof..
If both are zero, then is the identity matrix. If and , then let be the permutation matrix which swaps rows and .
Otherwise, let be as in Fact 1 with , and for . Define as the identity matrix except
Clearly satisfies the desired properties.
We note that off diagonal entries in a triangular matrix can be unimodularly reduced by the diagonal entry below it.
Lemma 3.
Let be upper triangular with nonzero diagonal. There exists a unimodular matrix , which is upper triangular and has ones on the diagonal, such that in every column of , the degree of each diagonal entry is strictly larger than the degrees of the entries above it.
Proof..
For any with , we have for quotient and remainder with , and
Embedding such unimodular matrices into identity matrices, we can “reduce” the off diagonal entries of by the diagonal entries below them.
Theorem 4.
Let have full rank. Then there exists a matrix in Hermite form, and a unimodular matrix , such that .
Proof..
The proof follows by observing the traditional (but inefficient) algorithm to compute the Hermite form. We first use a (unimodular row) permutation to move any nonzero element in column 1 into the top left position; failure to find a nonzero element in column 1 means our matrix is rank deficient. We then repeatedly apply Lemma 2 to find such that only has the top left position nonzero. This same procedure is then repeated on subdiagonal of columns in sequence, so there exists a unimodular matrix such that is upper triangular. The matrix is then unimodularly reduced using Lemma 3.
Theorem 5.
Let have full row rank. Suppose for unimodular and Hermite form . Then both and are unique.
Proof..
Suppose and are both Hermite forms of . Thus, there exist unimodular matrices and such that and , and where is unimodular. Since and are upper triangular matrices, we know is as well. Moreover, since and have monic diagonal entries, the diagonal entries of equal . We now prove is the identity matrix. By way of contradiction, first assume that is not the identity, so there exists an entry which is the first nonzero offdiagonal entry on the th row of . Since and since , . Because , we see , which contradicts the definition of the Hermite form.
Uniqueness of is easily established since , so and .
3 Noncommutative determinants and degree bounds for linear equations
One of the main difficulties in matrix computations in skew (noncommutative) fields, and a primary difference with the commutative case, is the lack of the usual determinant. In particular, the determinant allows us to bound the degrees of solutions to systems of equations, the size of the inverse or other decompositions, not to mention the degrees at intermediate steps of computations, through Hadamardlike formulas and Cramer’s rules.
The most common noncommutative determinant was defined by Dieudonné (1943), and is commonly called the Dieudonné determinant. It preserves some of the multiplicative properties of the usual commutative determinant, but is insufficient to establish the degree bounds we require (amongst other inadequacies). Gel’fand and Retakh (1991, 1992) introduced quasideterminants and a rich associated theory as a central tool in linear algebra over noncommutative rings. Quasideterminants are more akin to the (inverse of the) entries of the classical adjoint of a matrix than a true determinant. We employ quasideterminants here to establish bounds on the degree of the entries in the inverse of a matrix, and on the Dieudonné determinant in this section, and on the Hermite form and its multiplier matrices in Section 4.
We will establish bounds on degrees of quasideterminants and Dieudonné determinants for a general skew field with a degree satisfying the following properties. For :

If then , and ;

;

;

If then .
As a simple commutative example, if for some field and commuting indeterminate , for any with polynomials (), we can define .
More properly, our degree function is a nonarchimedean valuation on . Since our main application will be to noncommutative Ore polynomial rings, where degrees are a natural and traditional notion, we will adhere to the nomenclature of degrees. We note, however, that the degrees as defined here may become negative. See Lemma 13 for the effective application to the Ore polynomial case.
3.1 Quasideterminants and degree bounds
Following Gel’fand and Retakh (1991, 1992), we define the quasideterminant as a collection of functions from , where represents the function being undefined. Let and . Assume is the entry of , and let be the matrix with the th row and th column removed. Define the quasideterminant of as
where the sum is taken over all summands where is defined. If all summands have undefined then is undefined (and has value ). See (Gel’fand and Retakh, 1992).
Fact 6 (Gel’fand and Retakh (1991), Theorem 1.6).
Let over a (possibly skew) field .

The inverse matrix exists if and only if the following are true:

If the quasideterminant is defined then , for all ;

For all there exists a , such that the quasideterminant is defined;

For all there exists a such that the quasideterminant is defined;


If the inverse exists, then for we have
Over a commutative field , where has inverse , the quasideterminants behave like a classical adjoint: . If is zero then is undefined.
We now bound the size of the quasideterminants in terms of the size of the entries of . Assume that has a degree function as above.
Theorem 7.
Let , such that either or for all . For all such that is defined we have .
Proof. We proceed by induction on .
For , and , so clearly the property holds. Assume the statement is true for dimension . Then
where the sum is over all defined summands. Then using the inductive hypothesis we have
and
Corollary 8.
Let be unimodular, and such that . Assume or for all . Then .
Proof..
From Fact 6 we know that when is defined (and otherwise). Thus , and or since is unimodular.
3.2 Dieudonné Determinants
Let be the commutator subgroup of the multiplicative group of , the (normal) subgroup of generated by all pairs of elements of the form for . Thus is a commutative group.
Let be a matrix with a right inverse. The Bruhat Normal Form of is a decomposition , where is a permutation matrix inducing the permutation , and are
See (Draxl, 1983, Chapter 19) for more details. The Bruhat decomposition arises from Gaussian elimination, much as the decomposition does in the commutative case. We then define (sometimes called the predeterminant of ). Let be the canonical projection from . Then the Dieudonné determinant is defined as , or if is not invertible.
The Dieudonné determinant has a number of the desirable properties of the usual determinant, as proven in (Dieudonné, 1943):

for any ;

for any permutation matrix;

.
Also note that if has a degree function as above, then is well defined, since all elements of the equivalence class of have the same degree (since the degree of all members of the commutator subgroup is zero). Gel’fand and Retakh (1991) show that
when all these quasideterminants are defined (or equivalently is the identity in the Bruhat decomposition above), where is the matrix with rows and columns removed (keeping the original labelings of the remaining rows and columns).
More generally, let , be permutations of , let , , and define as the matrix with rows and columns removed (where ). Define
(3.1)  
Fact 9 (Gelfand et al. (2005), Section 3.1).
Let be permutations of and defined as above. If is defined for , then
In other words, the Dieudonné determinant is essentially invariant of the order of the sequence of submatrices specified in (3.1).
Theorem 10.
Let be invertible, with . Then .
Proof..
We proceed by induction on . For this is clear. For , the possible predeterminants are
at least one of which must be defined and nonzero, and all of which clearly have degree at most .
Now assume the theorem is true for matrices of dimension less than . Choose such that is nonzero and of minimal degree; that is for all such that is defined and nonzero. The fact that implies that is invertible, and we can continue this process recursively. Thus, let and be permutations of such that and is minimal over the degrees of nonzero, defined quasideterminants , for . Now
where all sums are taken only over defined quasideterminants as above. Thus
using the induction hypothesis and the assumption that is chosen to be minimal.
4 Degree bounds on matrices over
Some wellknown properties of are worth recalling; see (Ore, 1933) for the original theory or (Bronstein and Petkovšek, 1994) for an algorithmic presentation. Given , there is a degree function (in ) which satisfies the usual properties: and . We set .
is a left (and right) principal ideal ring, which implies the existence of a right (and left) division with remainder algorithm such that there exists unique such that where . This allows for a right (and left) Euclideanlike algorithm which shows the existence of a greatest common right divisor, , a polynomial of minimal degree (in ) such that and for . The GCRD is unique up to a left multiple in , and there exist cofactors such that . There also exists a least common left multiple . Analogously there exists a greatest common left divisor, , and least common right multiple, , both of which are unique up to a right multiple in . From (Ore, 1933) we also have that
(4.1)  
It will be useful to work in the quotient skew field of , and to extend the degree function appropriately. We first show that any element of can be written as a standard fraction , for (and in particular, since is noncommutative, we insist that is on the right).
Fact 11 (Ore (1933), Section 3).
Every element of can be written as a standard fraction.
The notion of degree extends naturally to as follows.
Definition 12.
For , , the degree .
The proof of the next lemma is left to the reader.
Lemma 13.
For , with , we have the following:

if then ;

;

;

.
In summary, the degree function on meets the requirement of a degree function on a skew field as in Section 3, and is once again, actually a valuation on .
4.1 Determinantal degree and unimodularity
We show unimodular matrices are precisely those with a Dieudonné determinant of degree zero.
Lemma 14.
Let be as in Fact 1. Then .
Proof..
We may assume that , since the same matrix satisfies. Also assume both (otherwise the lemma is trivial). Then
so . Since , from (4.1) we know , so .
Embedding the matrices into identity matrices, as in Lemma 2, we obtain the following (the proof of which is left to the reader).
Corollary 15.
Let be as in Lemma 2. The .
The characterization of unimodular matrices as those with Dieudonné determinant of degree zero follows by looking at the Hermite form of a unimodular matrix.
Theorem 16.
is unimodular if and only if .
Proof..
Suppose is unimodular. The Hermite form of must be the identity: all the diagonal entries must be invertible in and the entries above the diagonal are reduced to . Thus, the unimodular multiplier to the Hermite form of will be the inverse .
Following the simple algorithm to compute the Hermite form in Theorem 4, we see it worked via a sequence of unimodular transforms, all of which were either permutations, offdiagonal reductions from Lemma 2, or are of the form in Lemmas 2 and 15. The Dieudonné determinants of permutations and reduction transformations are both equal to , by the basic properties of Dieudonné determinants discussed at the beginning of Section 3.2, and hence have degree . The Dieudonné determinants of the transformations are of degree by Corollary 15. The proof is now complete by the multiplicative property of Dieudonné determinants, and the additive properties of their degrees.
Assume conversely that , and that is a unimodular matrix such that is in Hermite form. Then