On computing the Hermite form of a matrix of differential polynomials
aa r X i v : . [ c s . S C ] J un On computing the Hermite form of a matrix ofdifferential polynomials
Mark Giesbrecht and Myung Sub Kim
Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada
Abstract.
Given a matrix A ∈ F ( t )[ D ; δ ] n × n over the ring of differentialpolynomials, we show how to compute the Hermite form H of A and aunimodular matrix U such that UA = H . The algorithm requires apolynomial number of operations in F in terms of n , deg D A , deg t A .When F = Q it require time polynomial in the bit-length of the rationalcoefficients as well. Canonical forms of matrices over principal ideal domains (such as Z or F [ x ],for a field F ) have proven invaluable for both mathematical and computationalpurposes. One of the successes of computer algebra over the past three decadeshas been the development of fast algorithms for computing these canonical forms.These include triangular forms such as the Hermite form (Hermite, 1863), lowdegree forms like the Popov form (Popov, 1972), as well as the diagonal Smithform (Smith, 1861).Canonical forms of matrices over non-commutative domains, especially ringsof differential and difference operators, are also extremely useful. These havebeen examined at least since Dickson (1923), Wedderburn (1932), and Jacobson(1943). A typical domain under consideration is that of differential polynomials.For our purposes these are polynomials over a function field F ( t ) (where F is afield of characteristic zero, typically an extension of Q , or some representationof C ). A differential indeterminate D is adjoined to form the ring of differentialpolynomials F ( t )[ D ; δ ], which consists of the polynomials in F ( t )[ D ] under theusual addition and a non-commutative multiplication defined such that D a = a D + δ ( a ), for any a ∈ F ( t ). Here δ : F ( t ) → F ( t ) is a pseudo-derivative , a functionsuch that for all a, b ∈ F ( t ) we have δ ( a + b ) = δ ( a ) + δ ( b ) and δ ( ab ) = aδ ( b ) + δ ( a ) b. The most common derivation in F ( t ) takes δ ( a ) = a ′ for any a ∈ F ( t ), the usualderivative of a , though other derivations (say δ ( t ) = t ) are certainly of interest.A primary motivation in the definition of F ( t )[ D ; δ ] is that there is a nat-ural action on the space of infinitely differentiable functions in t , namely thedifferential polynomial a m D m + a m − D m − + · · · + a D + a ∈ F ( t )[ D ; δ ] acts as the linear differential operator a m ( t ) d m y ( t ) dt m + a m − ( t ) d m − y ( t ) dt m − + · · · + a ( t ) dy ( t ) dt + a ( t ) y ( t )on a differentiable function y ( t ). Solving and analyzing systems of such opera-tors involves working with matrices over F ( t )[ D ; δ ], and invariants such as thedifferential analogues of the Smith, Popov and Hermite forms provide importantstructural information.In commutative domains such as Z and F [ x ], it has been more common tocompute the triangular Hermite and diagonal Smith form (as well as the lowerdegree Popov form, especially as an intermediate computation). Indeed, theseforms are more canonical in the sense of being canonical in their class under mul-tiplication by unimodular matrices. Polynomial-time algorithms for the Smithand Hermite forms over F [ x ] were developed by Kannan (1985), with impor-tant advances by Kaltofen et al. (1987), Villard (1995), Mulders and Storjohann(2003), and many others. One of the key features of this recent work in com-puting normal forms has been a careful analysis of the complexity in terms ofmatrix size, entry degree, and coefficient swell. Clearly identifying and analyz-ing the cost in terms of all these parameters has led to a dramatic drop in boththeoretical and practical complexity.Computing the classical Smith and Hermite forms of matrices over differ-ential (and more general Ore) domains has received less attention though nor-mal forms of differential polynomial matrices have applications in solving dif-ferential systems and control theory. Abramov and Bronstein (2001) analyzesthe number of reduction steps necessary to compute a row-reduced form, whileBeckermann et al. (2006) analyze the complexity of row reduction in terms ofmatrix size, degree and the sizes of the coefficients of some shifts of the inputmatrix. Beckermann et al. (2006) demonstrates tight bounds on the degree andcoefficient 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 ap-proach involves order bases computation in order to eliminate lower order termsof Ore polynomial matrices. A main contribution of Cheng (2003) is to give analgorithm computing the row rank and a row-reduced basis of the left nullspaceof a matrix of Ore polynomials in a fraction-free way. This idea is extended inDavies et al. (2008) to compute Popov form of general Ore polynomial matrices.In Davies et al. (2008), they reduce the problem of computing Popov form to anullspace computation. However, though Popov form is useful for rewriting highorder terms with respect to low order terms, we want a different normal formmore suited to solving system of linear diophantine equations. Since the Hermiteform is upper triangular it meets this goal nicely, not to mention the fact thatit is a “classical” canonical form. In a slightly different vein, Middeke (2008)has recently given an algorithm for the Smith (diagonal) form of a matrix ofdifferential polynomials, which requires time polynomial in the matrix size anddegree (but the coefficient size is not analyzed).In this paper, we first discuss some basic operations with polynomials in F ( t )[ D ; δ ], which are typically written with respect to the differential variable D as f = f + f D + f D + · · · + f d D d , (1.1)where f , . . . , f d ∈ F ( t ), with f d = 0. We write d = deg D f to mean the degree inthe differential variable, and generally refer to this as the degree of f . Since this isa non-commutative ring, it is important to set a standard notation in which thecoefficients f , . . . , f d ∈ F ( t ) are written to the left of the differential variable D .For u, v ∈ F [ t ] relatively prime, we can define deg t ( u/v ) = max { deg t u, deg t v } .This is extended to f ∈ F ( t )[ D ; δ ] as in (1.1) by letting deg t f = max i { deg t f i } .We think of deg t as measuring coefficient size or height. Indeed, with a littleextra work the bounds and algorithms in this paper are effective over Q ( t ) aswell, where we also include the bit-length of rational coefficients, as well as thedegree in t , in our analyses.A matrix U ∈ F ( t )[ D ; δ ] n × n is said to be unimodular if there exists a V ∈ F ( t )[ D ; δ ] n × n such that U V = I , the n × n identity matrix. Note that we do notemploy the typical determinantal definition of a unimodular matrix, as thereis no easy notion of determinant for matrices over F ( t )[ D ; δ ] (indeed, workingaround this deficiency suffuses much of our work).A matrix H ∈ F ( t )[ D ; δ ] n × n is said to be in Hermite form if H is uppertriangular, if every diagonal entry is monic, and every off-diagonal entry hasdegree less than the diagonal entry below it. As an example, the matrix t + 2) D + D t + 1) D t ) D t + t + t D t + 2 t + D t + t t + (3 + t ) D + D t + (5 + 3 t ) D + D t + (2 + 4 t ) D has Hermite form t + D t − t +2 t t − t D t + D t + D − t + − t + t t D + D . Note that the Hermite form may have denominators in t . Also, while this exampledoes not demonstrate it, it is common that the degrees in the Hermite form, inboth t an D , are substantially larger than in the input.In this paper we will only concern ourselves with matrices in F ( t )[ D ; δ ] n × n of full row rank, that is, matrices whose rows are F ( t )[ D ; δ ]-linear independent.For any matrix A ∈ F ( t )[ D ; δ ] n × n , we show there exists a unimodular matrix U such that U A = H is in Hermite form. This form is canonical in the sensethat if two matrices A, B ∈ F ( t )[ D ; δ ] n × n are such that A = P B for unimodular P ∈ F ( t )[ D ; δ ] n × n then the Hermite form of A equals the Hermite form of B .The main contribution of this paper is an algorithm that, given a matrix A ∈ F ( t )[ D ; δ ] n × n (of full row rank), computes H and U such that U A = H ,which requires a polynomial number of F -operations in n , deg D A , and deg t A . Itwill also require time polynomial in the coefficient bit-length when F = Q .The remainder of the paper is organized as follows. In Section 2 we summarizesome basic properties of differential polynomial rings and present and analyze algorithms for some necessary basic operations. In Section 3 we introduce a newapproach to compute appropriate degree bounds on the coefficients of H and U . In Section 4 we present our algorithm for computing the Hermite form of amatrix of differential polynomials and analyze it completely. F [ t ][ D ; δ ] In this section we discuss some of the basic structure of the ring F ( t )[ D ; δ ] andpresent and analyze simple algorithms to do some computations that will benecessary in the next section.Some well-known properties of F ( t )[ D ; δ ] are worth recalling; seeBronstein and Petkovˇsek (1994) for an algorithmic presentation of this theory.Given f, g ∈ F ( t )[ D ; δ ], there is a degree function (in D ) which satisfies the usualproperties: deg D ( f g ) = deg D f + deg D g and deg D ( f + g ) ≤ max { deg D f, deg D g } . F ( t )[ D ; δ ] is also a left and right principal ideal ring, which implies the existenceof a right (and left) division with remainder algorithm such that there existsunique q, r ∈ F ( t )[ D ; δ ] such that f = qg + r where deg D ( r ) < deg D ( g ). Thisallows for a right (and left) euclidean-like algorithm which shows the existence ofa greatest common right divisor, h = gcrd( f, g ), a polynomial of minimal degree(in D ) such that f = uh and g = vh for u, v ∈ F ( t )[ D ; δ ]. The GCRD is uniqueup to a left multiple in F ( t ) \{ } , and there exist co-factors a, b ∈ F ( t )[ D ; δ ]such that af + bg = gcrd( f, g ). There also exists a least common left multiplelclm( f, g ). Analogously there exists a greatest common left divisor, gcld( f, g ),and least common right multiple, lcrm( f, g ), both of which are unique up to aright multiple in F ( t ).Efficient algorithms for computing products of polynomials are developed invan der Hoeven (2002) and Bostan et al. (2008), while fast algorithms to com-pute the LCLM and GCRD, are developed in Li and Nemes (1997) and Li (1998).In this paper we will only need to compute very specific products of the form D k f for some k ∈ N . We will work with differential polynomials in F [ t ][ D ; δ ], asopposed to F ( t )[ D ; δ ], and manage denominators separately. If f ∈ F [ t ][ D ; δ ] iswritten as in (1.1), then f , . . . , f d ∈ F [ t ], and D f = X ≤ i ≤ d f i D i +1 + X ≤ i ≤ d f ′ i D i ∈ F [ t ][ D ; δ ] , where f ′ i ∈ F [ t ] is the usual derivative of f i ∈ F [ t ]. Assume deg t f ≤ e . It is easilyseen that deg D ( D f ) = d + 1, and deg t ( D f ) ≤ e . The cost of computing D f is O ( de ) operations in F . Computing D k f , for 1 ≤ k ≤ m then requires O ( dem )operations in F .If F = Q we must account for the bit-length of the coefficients as well. Assum-ing our polynomials are in Z [ t ][ D ; δ ] (which will be sufficient), and are writtenas above, we have f i = P ≤ j ≤ e f ij t j for f ij ∈ Z . We write k f k ∞ = max | f ij | tocapture the coefficient size of f . It easily follows that kD f k ∞ ≤ ( e + 1) k f k ∞ ,and so kD m f k ∞ ≤ ( e + 1) m k f k ∞ . Lemma 2.1. (i) Let f ∈ F [ t ][ D ; δ ] have deg D f = d , deg t f = e , and let m ∈ N . Then we cancompute D k f , for ≤ k ≤ m , with O ( dem ) operations in F .(ii) Let f ∈ Z [ t ][ D ; δ ] . Then kD m f k ∞ ≤ ( e + 1) m · k f k ∞ , and we can compute D i f , for ≤ i ≤ m , with O ( dem · ( m log e + log k f k ∞ ) ) bit operations. We make no claim that the above methods are the most efficient, and fasterpolynomial and matrix arithmetic will certainly improve the cost. However, theabove analysis will be sufficient, and these costs will be dominated by others inthe algorithms of later sections.
In this section we prove the existence and uniqueness of the Hermite form over F ( t )[ D ; δ ], and prove some important properties about unimodular matrices andequivalence over this ring. The principal technical difficulty is that there is nonatural determinant function with the properties found in commutative linearalgebra. The determinant is one of the main tools used in the analysis of essen-tially all fast algorithms for computing the Hermite form H and transformationmatrix U , and specifically two relevant techniques in established methods byStorjohann (1994) and Kaltofen et al. (1987). One approach might be to employthe non-commutative determinant of Dieudonn´e (1943), but this adds consider-able complication. Instead, we find degree bounds via established bounds on therow-reduced form. Definition 3.1 (Unimodular matrix).
Let U ∈ F ( t )[ D ; δ ] n × n and supposethere exists a V ∈ F ( t )[ D ; δ ] n × n such that U V = I n , where I n is the identitymatrix over F ( t )[ D ; δ ] n × n . Then U is called a unimodular matrix over F ( t )[ D ; δ ] . This definition is in fact symmetric, in that V is also unimodular, as shown inthe following lemma (the proof of which is left to the reader). Lemma 3.1.
Let U ∈ F ( t )[ D ; δ ] n × n be unimodular such that there exists a V ∈ F ( t )[ D ; δ ] n × n with U V = I n . Then V U = I n as well. Theorem 3.1.
Let a, b ∈ F ( t )[ D ; δ ] . There exists a unimodular matrix W = (cid:18) u vs t (cid:19) ∈ F ( t )[ D ; δ ] × such that W (cid:18) ab (cid:19) = (cid:18) g (cid:19) , where g = gcrd( a, b ) and sa = − tb = lclm( a, b ) .Proof. Let u, v ∈ F ( t )[ D ; δ ] be the multipliers from the euclidean algorithm suchthat ua + vb = g . Since sa = − tb = lclm( a, b ), we know that gcld( s, t ) = 1(otherwise the minimality of the degree of the lclm would be violated). It followsthat there exist c, d ∈ F ( t )[ D ; δ ] such that sc + td = 1. Now observe that (cid:18) u vs t (cid:19) (cid:18) ag − cbg − d (cid:19) (cid:18) − uc − vd (cid:19) = (cid:18) uc + vd (cid:19) (cid:18) − uc − vd (cid:19) = (cid:18) (cid:19) . Thus W − = (cid:18) ag − ag − ( − uc − vd ) + cbg − bg − ( − uc − vd ) + d (cid:19) = (cid:18) ag − − a + cbg − − b + d (cid:19) , so W is unimodular. ⊓⊔ Definition 3.2 (Hermite Normal Form).
Let H ∈ F ( t )[ D ; δ ] n × n with fullrow rank. The matrix H is in Hermite form if H is upper triangular, if everydiagonal entry of H is monic, and if every off-diagonal entry of H has degree(in D ) strictly lower than the degree of the diagonal entry below it. Theorem 3.2.
Let A ∈ F ( t )[ D ; δ ] n × n have row rank n . Then there exists amatrix H ∈ F ( t )[ D ; δ ] n × n with row rank n in Hermite form, and a unimodularmatrix U ∈ F ( t )[ D ; δ ] n × n , such that U A = H .Proof. We show this induction on n . The base case, n = 1, is trivial and wesuppose that the theorem holds for n − × n − A has row rank n , we can find a permutation of the rows of A such that every principal minor of A has full row rank. Since this permutation is a unimodular transformation of A , we assume this property about A . Thus, by the induction hypothesis, thereexists a unimodular matrix U ∈ F ( t )[ D ; δ ] ( n − × ( n − such that U · · · · A = ¯ H = ¯ H , · · · · · · ∗ ∗ ¯ H , · · · ∗ ∗ . . . ... ...¯ H n − ,n − ∗ A n, A n, · · · A n,n − A n,n ∈ F ( t )[ D ; δ ] n × n , where the ( n − H is in Hermite form. By Theorem 3.1,we know that there exists a unimodular matrix W = (cid:18) u i v i s i − t i (cid:19) ∈ F ( t )[ D ; δ ] × such that W (cid:18) ¯ H ii A n,i (cid:19) = (cid:18) g i (cid:19) ∈ F ( t )[ D ; δ ] × . This allows us to reduce A n, , . . . , A n,n − to zero, and does not introduce anynon-zero entries below the diagonal. Also, all off-diagonal entries can be reducedusing unimodular operations modulo the diagonal entry, putting the matrix intoHermite form. ⊓⊔ Corollary 3.1.
Let A ∈ F ( t )[ D ; δ ] n × n have full row rank. Suppose U A = H forunimodular U ∈ F ( t )[ D ; δ ] n × n and Hermite form H ∈ F ( t )[ D ; δ ] n × n . Then both U and H are unique.Proof. Suppose H and G are both Hermite forms of A . Thus, there exist uni-modular matrices U and V such that U A = H and V A = G , and G = W H where W = V U − is unimodular. Since G and H are upper triangular matrices,we know W is as well. Moreover, since G and H have monic diagonal entries,the diagonal entries of W equal 1. We now prove W is the identity matrix. By way of contradiction, first assume that W is not the identity, so there exists anentry W ij which is the first nonzero off-diagonal entry on the i th row of W .Since i < j and since W ii = 1, G ij = H ij + W ij H jj . Because W ij = 0, we seedeg D G ij ≥ deg D G jj , which contradicts the definition of the Hermite form. Theuniqueness of U follows similarly. ⊓⊔ Definition 3.3 (Row Degree).
A matrix T ∈ F ( t )[ D ; δ ] n × n has row degree −→ u ∈ ( N ∪ {−∞} ) n if the i th row of T has degree u i . We write rowdeg −→ u . Definition 3.4 (Leading Row Coefficient Matrix).
Let T ∈ F ( t )[ D ; δ ] n × n have rowdeg −→ u . Set N = deg D T and S = diag( D N − u , . . . , D N − u n ) . We write ST = L D N + lower degree terms in D , where the matrix L = LC row ( T ) ∈ F ( t ) n × n is called the leading row coefficientmatrix of T . Definition 3.5 (Row-reduced Form).
A matrix T ∈ F ( t )[ D ; δ ] m × s with rank r is in row-reduced form if rank LC row ( T ) = r . Fact 3.1 (Beckermann et al. (2006) Theorem 2.2).
For any A ∈ F ( t )[ D ; δ ] m × s there exists a unimodular matrix U ∈ F ( t )[ D ; δ ] m × m , with T = U A having r ≤ min { m, s } nonzero rows, rowdegT ≤ rowdegA , and where the submatrixconsisting of the r nonzero rows of T are row-reduced. Moreover, the unimodularmultiplier satisfies the degree bound rowdegU ≤ −→ v + ( |−→ u | − |−→ v | − min j { u j } ) −→ e , where −→ u := max( −→ , rowdegA ) , −→ v := max( −→ , rowdegT ) , and −→ e is the columnvector with all entries equal to 1. The proof of the following is left to the reader.
Corollary 3.2. If A ∈ F ( t )[ D ; δ ] n × n is a unimodular matrix then the row re-duced form of A is an identity matrix. The following theorems provide degree bounds on H and U . We first computea degree bound of the inverse of U by using the idea of backward substitution,and then use the result of Beckermann et al. (2006) to compute degree boundof U . Theorem 3.3.
Let A ∈ F ( t )[ D ; δ ] n × n be a matrix with deg D A ij ≤ d and fullrow rank. Suppose U A = H for unimodular matrix U ∈ F ( t )[ D ; δ ] n × n and H ∈ F ( t )[ D ; δ ] n × n in Hermite form. Then there exist a unimodular matrix V ∈ F ( t )[ D ; δ ] n × n such that A = V H where
U V = I n and deg D V ij ≤ d .Proof. We prove by induction on n . The base case is n = 1. Since H =gcrd( A , . . . , A n ), deg D H ≤ d and so deg D V i ≤ d for 1 ≤ i ≤ n . Now, we suppose that our claim is true for k where 1 < k < n . Then we have to showthat deg D V ik +1 ≤ d . We need to consider two cases:Case 1: deg D V i,k +1 > max(deg D V i , . . . , deg D V ik ). Sincedeg D H k +1 ,k +1 ≥ max(deg D H ,k +1 , . . . , deg D H k,k +1 ) , deg D A i,k +1 = deg D ( V i,k +1 H k +1 ,k +1 ) , where A i,k +1 = V i H ,k +1 + · · · + V i,k +1 H k +1 ,k +1 . Thus, deg D V i,k +1 ≤ d .Case 2: deg D V i,k +1 ≤ max(deg D V i , . . . , deg D V ik ). Thus, by induction hypothe-sis, deg D V i,k +1 ≤ d . ⊓⊔ Corollary 3.3.
Let A , V , and U be those in Theorem 3.3. Then deg D U ij ≤ ( n − d .Proof. By Corollary 3.2, we know that the row reduced form of V is I n . Moreover,since I n = U V , we can compute the degree bound of U by using Fact 3.1. Clearly, −→ v + ( |−→ u | − |−→ v | − min j { u j } ) −→ e ≤ −→ v + ( |−→ u | − min j { u j } ) −→ e , where −→ u := max( −→ , rowdegV ) and −→ v := max( −→ , rowdegI n ) = −→ V is bounded by d , ( |−→ u | − min j { u j } ) ≤ ( n − d . Then,by Fact 3.1, rowdegU ≤ ( n − d . Therefore, deg D U ij ≤ ( n − d . ⊓⊔ Corollary 3.4.
Let H be same as that in Theorem 3.3. Then deg D H ij ≤ nd .Proof. Since deg D U ij ≤ ( n − d and deg D A ij ≤ d , deg D H ij ≤ nd . ⊓⊔ F ( t ) In this section we present our polynomial-time algorithm to compute the Hermiteform of a matrix over F ( t )[ D ; δ ]. We exhibit a variant of the linear system methoddeveloped in Kaltofen et al. (1987) and Storjohann (1994). The approach of thesepapers is to reduce the problem of computing the Hermite of matrices with(usual) polynomial entries in F [ z ] to the problem of solving a linear systemequations over F . Analogously, we reduce the problem of computing the Hermiteform over F [ t ][ D ; δ ] to solving linear systems over F ( t ). The point is that the field F ( t ) over which we solve is the usual, commutative, field of rational functions.For convenience, we assume that our matrix is over F [ t ][ D ; δ ] instead of F ( t )[ D ; δ ], which can easily be achieved by clearing denominators with a “scalar”multiple from F [ t ]. This is clearly a unimodular operation in the class of matricesover F ( t )[ D ; δ ].We first consider formulating the computation of the Hermite form a matrixover F ( t )[ D ; δ ] as the solution of a “pseudo”-linear system over F ( t )[ D ; δ ] (i.e., amatrix equation over the non-commutative ring F ( t )[ D ; δ ]). Theorem 4.1.
Let A ∈ F [ t ][ D ; δ ] n × n have full row rank, with deg D A i,j ≤ d ,and ( d , . . . , d n ) ∈ N n be given. Consider the system of equations P A = G , for n × n matrices for P, G ∈ F ( t )[ D ; δ ] restricted as follows: – The degree (in D ) of each entry of P is bounded by ( n − d + max ≤ i ≤ n d i . – The matrix G is upper triangular, where every diagonal entry is monic andthe degree of each off-diagonal entry is less than the degree of the diagonalentry below it. – The degree of the i th diagonal entry of G is d i .Let H be the Hermite form of A and ( h , . . . , h n ) ∈ N n be the degrees of thediagonal entries of H . Then the following are true:(a) There exists at least one pair P, G as above with
P A = G if and only if d i ≥ h i for ≤ i ≤ n .(b) If d i = h i for ≤ i ≤ n then G is the Hermite form of A and P is aunimodular matrix.Proof. The proof is similar to that of Kaltofen et al. (1987), Lemma 2.1. Givena degree vector ( d , . . . , d n ), we view P A = G as a system of equations in theunknown entries of P and G . Since H is the Hermite form of A , there exist aunimodular matrix U such that U A = H . Thus P U − H = G and the matrix P U − must be upper triangular since the matrices H and G are upper triangular.Moreover, since the matrix P U − is in F ( t )[ D ; δ ] n × n , and G ii = ( P U − ) ii · H ii for 1 ≤ i ≤ n , we know d i ≥ h i for 1 ≤ i ≤ n . For the other direction, wesuppose d i ≥ h i for 1 ≤ i ≤ n . Let D = diag( D d − h , . . . , D d n − h n ). Then since( DU ) A = ( DH ), we can set P = DU and G = DH as a solution to P A = G ,and the i th diagonal of G has degree d i by construction. By Corollary 3.3, weknow deg D U i,j ≤ ( n − d and so deg D P i,j ≤ ( n − d + max ≤ i ≤ n d i .To prove (b), suppose d i = h i for 1 ≤ i ≤ n and that, contrarily, G is not theHermite form of A . Since P U − is an upper triangular matrix with ones on thediagonal, P U − is a unimodular matrix. Thus P is a unimodular matrix and,by Corollary 3.1, G is the (unique) Hermite form of A , a contradiction. ⊓⊔ Lemma 4.1.
Let A , P , ( d , . . . , d n ) , and G be as in Theorem 4.1, and let β := ( n − d + max ≤ i ≤ n d i . Also, assume that deg t A ij ≤ e for ≤ i, j ≤ n .Then we can express the system P A = G as a linear system over F ( t ) as b P b A = b G where b P ∈ F ( t ) n × n ( β +1) , b A ∈ F [ t ] n ( β +1) × n ( β + d +1) , b G ∈ F ( t ) n × n ( β + d +1) . Assuming the entries b A are known while the entries of b P and b G are indeter-minates, the system of equations from b P b A = b G for the entries of b P and b G islinear over F ( t ) in its unknowns, and the number of equations and unknowns is O ( n d ) . The entries in b A are in F [ t ] and have degree at most e .Proof. Since deg D P i,j ≤ β , each entry of P has at most ( β +1) coefficients in F ( t )and can be written as P ij = P ≤ k ≤ β P ijk D k . We let b P ∈ F ( t ) n × n ( β +1) be thematrix formed from P with P ij replaced by the row vector ( P ij , . . . , P ijβ ) ∈ F ( t ).Since deg D P ≤ β , when forming P A , the entries in A are multiplied by D ℓ for0 ≤ ℓ ≤ β , resulting in polynomials of degree in D of degree at most µ = β + d . Thus, we construct b A as the matrix formed from A with A ij replaced by the( β + 1) × ( µ + 1) matrix whose ℓ th row is( A [ ℓ ] ij , A [ ℓ ] ij , . . . , A [ ℓ ] ijµ ) such that D ℓ A ij = A [ ℓ ] ij + A [ ℓ ] ij D + · · · + A [ ℓ ] ijµ D µ . Note that by Lemma 2.1 we can compute D ℓ A i,j quickly.Finally, we construct the matrix b G . Each entry of G has degree in D of degreeat most nd ≤ n ( β + d + 1). Thus, initially b G is the matrix formed by G with G ij replaced by( G ij , . . . , G ijµ ) where G ij = G ij + G ij D + · · · + G ijµ D µ . However, because of the structure of the system we can fix values of many ofthe entries of b G as follows. First, since every diagonal entry of the Hermite formis monic, we know the corresponding entry in b G is 1. Also, by Corollary 3.4,the degree in D of every diagonal entry of H is bounded by nd , and every off-diagonal has degree in D less than that of the diagonal below it (and hence lessthan nd ), and we can set all coefficients of larger powers of D to 0 in b G .The resulting system b P b A = b G , restricted as above according to Theorem 4.1,has O ( n d ) linear equations in O ( n d ) unknowns. Since the coefficients in b A areall of the form D ℓ A ij , and since this does not affect their degree in t , the degreein t of entries of b A is the same as that of A , namely e . ⊓⊔ With more work, we believe the dimension of the system can be reducedto O ( n d ) × O ( n d ) if we apply the techniques presented in Storjohann (1994)Section 4.3, wherein the unknown coefficients of b G are removed from the system.See also Labhalla et al. (1996).So far, we have shown how to convert the differential system over F ( t )[ D ; δ ]into a linear system over F ( t ). Also, we note, by Theorem 4.1, that the correctdegree of the i th diagonal entry in the Hermite form of A can be found byseeking the smallest non-negative integer k such that P A = G is consistentwhen deg D G j,j = nd for j = 1 , . . . , i − , i + 1 , . . . , n and k ≤ deg D G i,i . Usingbinary search, we can find the correct degrees of all diagonal entries by solvingat most O ( n log( nd )) systems. We then find the correct degrees of the diagonalentries in the Hermite form of A , solving the system P A = G with the correctdiagonal degrees gives the matrices U and H such that U A = H where H is theHermite form of A . Theorem 4.2.
Let A ∈ F [ t ][ D ; δ ] n × n with deg D A ij ≤ d and deg t A ij ≤ e for ≤ i, j ≤ n . Then we can compute the Hermite form H ∈ F ( t )[ D ; δ ] of A , and aunimodular U ∈ F [ t ][ D ; δ ] such that U A = H , with O (( n d + n d e ) log( nd )) operations in F Proof.
Lemma 4.1 and the following discussion, above shows that computing U and H is reduced to solving O ( n log( nd )) systems of linear equations over F ( t ),each of which is m × m for m = O ( n d ) and in which the entries have degree e . Using standard linear algebra this can be solved with O ( m e ) operations in F , since any solution has degree at most me (see von zur Gathen and Gerhard (2003)). A somewhat better strategy is to use the t -adic lifting approach of Dixon(1982), which would require O ( m + m e ) operations in F for each system, givinga total cost of O (( n d + n d e ) log( nd )) operations in F . ⊓⊔ As noted above, it is expected that we can bring this cost down througha smaller system similar to that of Storjohann (1994), to a cost of O (( n d + n d e ) log( nd )). Nonetheless, the algorithm as it is stated achieves a guaranteedpolynomial-time solution.It is often the case that we are considering differential systems over Q ( t )[ D ; δ ],where we must contend with growth in coefficients in D , t and in the size ofthe rational coefficients. However, once again we may employ the fact that theHermite form and unimodular transformation matrix are solutions of a linearsystem over Q [ t ]. For convenience, we can assume in fact that our input is in Z [ t ][ D ; δ ] n × n (since the rational matrix to eliminate denominators is unimodularin Q ( t )[ D ; δ ]). There is some amount of extra coefficient growth when going from A to b A ; namely we take up to nd derivatives, introducing a multiplicative con-stant of size around min(( nd )! , e !). In terms of the bit-length of the coefficients,this incurs a multiplicative blow-up of only O ( ℓ log( ℓ )) where ℓ = min( nd, e ). Itfollows that we can find the Hermite form of A ∈ Q ( t )[ D ; δ ] n × n in time polyno-mial in n , deg t A ij , deg D A ij , and log k A ij k , the maximum coefficient length inan entry, for 1 ≤ i, j ≤ n . A modular algorithm, for example along the lines ofLi and Nemes (1997), would improve performance considerably, as might p -adicsolvers and a more careful construction of the linear system. We have shown that the problem of computing the Hermite form of a matrix over F ( t )[ D ; δ ] can be accomplished in polynomial time. Moreover, our algorithm willalso control growth in coefficient bit-length when F = Q . We have also shownthat the degree bounds on Hermite forms in the differential ring are very similarto the regular polynomial case. From a practical point of view our method isstill expensive. Our next work will be to investigate more efficient algorithms.We have suggested ways to compress the system of equations and to employstructured matrix techniques. Also, the use of randomization has been shown tobe highly beneficial over F [ t ], and should be investigated in this domain. Finally,our approach should be applicable to difference polynomials and more generalOre polynomial rings. References
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