Determinant Formulas for Inhomogeneous Linear Differential, Difference and q-difference Equations

MM Research Preprints, 112 119 No. 19, Dec. 2000. Beijing Determinant Formulas for Inhomogeneous Linear Differential, Difference and q-difference Equations Ziming Li MMRC, Academy of Mathematics and Systems Science Chinese Academy of Sciences, Beijing 100080, China Abstract. This paper describes the gap-structure of subresultants in an Ore polynomial module, and the geometric explanation of isolated subresultants, which are peculiar to inhomogenous equations. As an application of the gap-structure, we present three determinant formulas extending the determinant formulas for the greatest common right divisor (gcrd), least common left multiple (lclm), and Sylvester s resultant of two elements in an Ore polynomial ring. 1. Introduction Determinant formulas are useful in usual commutative polynomial algebras. Well-known examples are the Sylvester resultant and subresultants of two univariate polynomials [?,?,?]. Besides their theoretical interests, these determinant formulas are also useful in algorithmic aspects. For example, univariate subresultants describe the coefficient growth of polynomial remainder sequences of two polynomials, and the Sylvester resultants help us to prove that there are only a finite number of unlucky primes in modular gcd-algorithms. Ore polynomial rings [?,?] establish a general setting for linear homogeneous differential, difference and q-difference polynomials. A subresultant theory for Ore polynomials [?,?] is developed to manipulate Ore polynomials. The subresultant algorithm and determinant formulas for gcrd s, lclm s and resultants are useful to design efficient modular gcrd-algorithms, fraction-free lclm-algorithms, and to estimate the complexity of these algorithms. Ore polynomial modules extend Ore polynomial rings to describe linear inhomogeneous differential, difference and q-difference polynomials in one framework [?], in which the gapstructure of subresultants in an Ore polynomial module is given as a corollary of the subresultant theorem. But the proof of the subresultant theorem is very technical because we need to find precise recursive relations among leading coefficients of non-zero subresultants, whereas, the gap-structure has nothing to do with these complicated formulas. To reveal the simplicity of the gap-structure, we give a direct proof. It is worthwhile because the gap-structure has many interesting applications. We wish that the direct proof would split the subresultant theory [?] into two parts: a simple part describing the gap-structure of a subresultant chain; and a technical part calculating the linear dependence among subresultants precisely. Becasue of the presence of inhomogeneous components, the gap-structure of the subresultant chain of two elements in an Ore polynomial module may contain an isolated suresultant, which is defective and is linear independent with any regular subresultant. The presence of isolated subresultants is due to the fact that the solution space of an inhomogeneous linear operational equation is no logner a vector space (see Corollary??). We also

Determinants for Inhomogeneous Equations 113 generalize the notion of gcrd s, lclm s and resultants in Ore polynomial rings to Ore polynomial modules. The major difficulty for the generalization is that an Ore submodule may not be generated by one element. Recall that a left ideal of an Ore polynomial ring is always principal. We find that a right replacement of a gcrd of two elements in an Ore polynomial module is their last nonzero subresultant (see Propositions?? and??). The paper is organized as follows. Section?? recalls the definition of Ore polynomial modules. Section?? defines subresultants in an Ore polynomial module. Sesction?? is devoted to describing the gap-structure of a subresultant chain. Section?? presents some applications of the gap-structure. 2. Ore polynomial modules In this section we recall the definition of Ore polynomial modules. Because of the limit of space, the proofs of assertions made in the section are omitted. These proofs are all quite easy. The reader is refered to [?] for details. Throught the paper, let F be a commutative field and F [X] an Ore polynomial ring with conjugate operator σ and pseudo-derivation δ. Consider the left linear space F [X] F over F. We define the degree of an element A a of F [X] F to be the degree of A if A is nonzero. For nonzero a F, the degree of 0 a is set to be 1. The element 0 0 is of degree and denoted by 0 for brevity. An element A a of F [X] F is understood as where m = deg A a and the a s are in F. A a = a m X m + + a 1 X + a 0 X 0 + ax 1, Definition 2.1 An endomorphism Θ of the additive group F [X] F is said to be an Ore operator on F [X] F if the following hold: 1. Θ(X n ) = X n+1, for n N. 2. For every a F, deg Θ(aX 1 ) 1. 3. (Multiplicative rule) For every r F and A a F [X] F, Θ(r(A a)) = σ(r)θ(a a) + δ(r)(a a). (1) The quadruple (F [X] F, Θ, σ, δ) is called an Ore polynomial module. Example 2.2 Let F be C(t) and D = d dt. Define the Ore operator Θ on F [X] F to be such that Θ(rX n ) = rx n+1 + D(r)X n and Θ(rX 1 ) = D(r)X 1, for all r F and n N. The F -module (F [X] F, Θ, 1, D) models the C(t)-linear differential space consisting of inhomogeneous linear differential equations in one unknown. Example 2.3 Let F be Q(t) and E be the shift operator on F sending t = t + 1. Define the Ore operator Θ on F [X] F to be such that Θ(rX n ) = E(r)X n+1 and Θ(rX 1 ) = E(r)X 1, for all r F and n N. (F [X] F, Θ, E, 0) models the Q(t)-module of inhomogeneous shift equations in one unknown.

114 Z. Li In the rest of this paper, (F [X] F, Θ, σ, δ) is assumed to be an Ore polynomial module and simply denoted by F [X] F. If a submodule M of F [X] F has the property that Θ(M) M, then M is called an Ore submodule. If N is a subset of F [X] F, then the multiplicative rule (??) implies that the smallest Ore submodule containing N is the submodule [N ] generated by all the elements of Θ(N ), which we call the Θ-submodule generated by N. Two elements A and B of F [X] F are said to be compatible if [A, B] does not contain any element of degree 1. Let A and B be in F [X] F, with respective degrees m and n, where n 0. A remainder of A and B is defined to be either A, if m < n; or C F [X] F such that deg C < deg B and A = m n i=0 r iθ i B + C, where r i belongs to F, for i = 0, 1,..., m n. The remainder just defined can be computed by a process analogous to the algebraic division. As deg(θ i+1 B) = deg(θ i B) + 1, for all i N, the remainder of A and B is unique. We denote the remainder of A and B by rem(a, B). For A and B in F [X] F, define that A and B are similar if either both A and B are zero or A and B are nonzero and linearly dependent over F. The similarity is an equivalence relation and denoted by. For A 1, A 2 F [X] F with deg(a 1 ) deg(a 2 ) 0, let A 1, A 2,..., A k be a sequence of nonzero elements of F [X] F such that A i rem(a i 2, A i 1 ), for i = 3,... k, and either deg(a k ) = 1 or rem(a k 1, A k ) = 0. Such a sequence is called a polynomial remainder sequence (p.r.s.) of A 1 and A 2. Just as in the algebraic case, A 1 and A 2 are compatible if and only if deg(a k ) 0. Because of the existence of the Euclidean algorithm, an Ore submodule not containing any element of degree 1 can always be generated by one element. Hence, [A 1, A 2 ] is generated by one element if and only if A 1 and A 2 is compatible, and [A 1 ] [A 2 ] is always generated by one element. 3. Subresultants In this section, we define the subresultants of elements in an Ore polynomial module. Since an element of F [X] F may have an inhomogeneous component we need to extend the notion of determinant polynomials [?,?]. Definition 3.1 Let M be an r c matrix with entries in F. If r c, then the extended determinant polynomial of M is defined to be M = c r 1 i= 1 det(m i )X i, (2) where M i is the r r matrix whose first (r 1) columns are the first (r 1) columns of M and whose last column is the (c i 1)th column of M, for i = 1, 0,..., c r 1. The extended determinant polynomial of M just defined is the determinant polynomial of M multiplied by X 1 from the right-hand side. Note that M is (left) multi-linear and alternative w.r.t. its rows. Let A : A 1, A 2,..., A m be a sequence in F [X] F. We denote by deg A the maximum of the degrees of the members in A. Let deg A = n > 1. The matrix associated with A is defined to be the m (n + 2) matrix whose entry in the ith row and jth column is the coefficient of X n+1 j in A i, for i = 1,..., m, and j = 1,..., n + 2. This matrix is denoted

Determinants for Inhomogeneous Equations 115 by mat(a). If m n + 2, then the extended determinant polynomial of A is defined to be mat(a), which is simply denoted by A. Remark 3.2 Notice that A = m k=1 r k A k, where r k F. If A is nonzero, then A 1,..., A k are F -linearly independent. The proof is not difficult [?]. Definition 3.3 Let A and B be in F [X] F with respective degrees m and n, where m n 0. For n 1 j 1, the jth subresultant of A and B is defined to be sres j (A, B) = Θ n j 1 A,..., ΘA, A, Θ m j 1 B,..., ΘB, B, }{{}}{{} n j m j The nth subresultant of A and B is defined to be B. The sequence S(A, B) : A, B, sres n 1 (A, B),..., sres 1 (A, B) is called the subresultant sequence of A and B. The next lemma [?,?] records some properties of subresultants. Lemma 3.1 If A and B are in F [X] F with respective degrees m and n, where m n 0, then 1. sres j (A, B) [A, B], where n 1 j 1; 2. deg(sres j (A, B)) j, where n 1 j 1; 3. sres n 1 (A, B) rem(a, B). 4. Gap-structure of a subresultant sequence Notation To avoid endlessly repeating the same assumptions, in the rest of the paper we let A and B be in F [X] F, with respective degrees m and n, where m n 0. Let S n be B and S j be sres j (A, B), for j = n 1, n 2,..., 1. The subresultant sequence S(A, B) consists of S n+1 = A, S n = B, S n 1,..., S 0, S 1. The proof of the following lemma is the same as that of Lemmas 4.2 in [?]. Lemma 4.1 If B H and S n 1 (A, B) G, then, for n 1 j 1, S j Θ m j 1 H,..., H, Θ n j 1 G,..., G. (3) If deg S j = j, then S j is regular. Otherwise S j is defective. In particular, the nth subresultant S n is always regular. The next theorem is the main result. Theorem 4.2 If S j+1 is regular and S j has degree r, for some j with n 1 j 0, then the following hold: 1. If r 1, then S i = 0, (j 1 i 1). (4) 2. If r 0, then and S i = 0, (j 1 i r + 1), (5) S r S j, (6) S r 1 rem(s j+1, S j ). (7)

116 Z. Li Proof We proceed by induction on n, the degree of S n. If n = 0, then S 1 rem(s 1, S 0 ) by Lemma??. The theorem is true since i can only be 1. Suppose that the theorem holds if deg B < n. Consider the case deg B = n. Since S n 1 rem(s n+1, S n ) by Lemma??, Lemma?? implies that, for all i with n 1 i 1, S i R i, (8) where R i = Θ m 1 i S n,..., S n, Θ n 1 i S n 1,..., ΘS n 1, S n 1. There are two cases. First, j = n 1. If r 1, then the degrees of both ΘS n 1 and S n 1 are of degree 1. It follows from (??) that S i = 0 for n 2 i 1. Equation (??) holds for j = n 1. Assume now that r > 1. If n 2 i r +1, then deg S n > 1 + deg Θ n 1 i S n 1. Thus, S i = 0 by (??). If i = r, then deg S n = 1 + deg Θ n 1 i S n 1. Thus, R r S n 1, so that S r S n 1 by (??). If i = r 1 then deg S n = deg Θ n 1 i S n 1. Hence, R i sres r 1 (S n, S n 1 ), so S i sres r 1 (S n, S n 1 ) by (??) and S i rem(b, S n 1 ) by Lemma??. The theorem holds for j = n 1. It remains to consider the case j < n 1. Actually, j < r since S r is the regular subresultant next to S n. Equation (??) and the fact S n 1 S r just proved in the last paragraph imply that S i sres i (S n, S r ), for i = r 1, r 2,..., 1 because deg Θ r 1 i S n = Θ n 1 i S n 1. The theorem then follows from the induction hypothesis on the subresultant chain of S n and S r : S n, S r, sres r 1 (S n, S r ),..., sres 0 (S n, S r ), sres 1 (S n, S r ). The next definition is particular for elements in an Ore polynomial module. Definition 4.1 A defective subresultant of degree 1 is said to be isolated. The gap structure of S(A, B) is given in Figure 1. A B... Sj+1 is regular. S j is defective of degree r.. S i = 0 (j > i > r). S r is regular.... a regular subresultant a defective subresultant. zero subresultants a regular subresultant an isolated subres. if one exists. Fig. 1. The gap structure of S(A, B) Note that the gap-structure of S(A, B) is slightly more complicated than that of an algebraic subresultant sequence due to the possible presence of isolated subresultants.

Determinants for Inhomogeneous Equations 117 Now, we extend subresultant sequences of the first and second kinds and prove that subresultant sequences of the first kind are p.r.s s. Definition 4.2 The subresultant sequence of A and B of the first kind is the subsequence of S(A, B) that consists of A, B, and S j if S j+1 is regular and S j is nonzero. The subresultant sequence of A and B of the second kind is the subsequence of S(A, B) that consists of A, B and other regular subresultants of S(A, B). The subresultant sequences of A and B of the first and second kinds are denoted by S 1 (A, B) and S 2 (A, B), respectively. At last, we prove Theorem 4.3 S 1 (A, B) is a p.r.s.. Proof Set A = A 1 and B = A 2. Let A 1, A 2,..., A p be a p.r.s.. Let S 1 (A, B) consist of A 1 A 2 S j3,..., S jq. We show that p = q and S ji A i, for i = 3,..., p. First, we consider the special case p = 2. Since rem(a 1, A 2 ) = 0, S n 1 = 0 by Lemma??. Theorem?? then implies that S j = 0, for all j < n, so that S 1 (A, B) consists of A 1, A 2. The theorem is true. Second, we consider the case rem(a 1, A 2 ) 0. Since j 3 = n 1 by the definition of S 1 (A, B), S j3 = S n 1 A 3 by Lemma??. Assume that A i S ji, for all i < k. Since S jk rem ( ) S jk 1 +1, S jk 1 by (??), Sjk 1 +1 S jk 2 by (??), and S jk 2 A k 2 by the induction hypothesis, we conclude S jk A k. We have proved that S ji A i, for i = 3,..., p. It remains to show that q = p. If deg A p = 1, then deg S p = 1. Thus, S i = 0 for i < p, so p = q. Suppose now that rem(a p 1, A p ) = 0. Let d = deg A p. Applying Theorem?? to (S jp 1 +1, S jp 1 ) and (S jp+1, S jp ), respectively, we derive S jp 1 S jp+1, S i = 0 for j p 1 i d+1, and S d S jp. Since rem(s jp 1, S jp ) = 0, rem(s jp+1, S jp ) = 0. It follows from (??) that S d 1 = 0 Applying Theorem?? to (S d, S d 1 ) yields S i = 0 for d 2 i 1. Hence, S jp is a last member of S 1 (A, B). 5. Applications In addition to the notation convention made in the last section, we denote the last nonzero elements of S(A, B) by S d. First, we study the Ore submodule [A] + [B]. Proposition 5.1 If deg S d 0, then [A, B] = [S d ]. Otherwise, A and B are not compatible. Proof Denote by S l the last member of S 1 (A, B). If deg S d 0, then deg S l 0, so [A, B] = [S l ]. Hence, [A, B] = [S d ] since S d S l by Theorem?? (applied to (S l 1, S l )). If deg S d = 1, then d = l. It follows that A and B are not compatible. From Proposition?? we see that S d is a proper extension of the notion of gcrds when deg S d 0. If deg S d = 1, then [A, B] is generated by two elements S l 1 and S l. Next, we extend the notion of resultants to Ore polynomial modules. Corollary 5.2 Denote by r d the coefficient of X d in S d If d = 1, then [A, B] contains an element with degree 1. Assume that d 0. Then [A, B] is generated by S d if and only if r d 0.

118 Z. Li Proof If d = 1, then S 1 [A, B]. Suppose that d 0. If r d 0, then deg S d = d 0, so [A, B] = [S d ] by Proposition??. If r d = 0, then S d is isolated, for, otherwise, S d would not be the last nonzero one by Theorem??. Hence, A and B are not compatible. At last, we study [A] [B]. Since the degrees of A and B are both non-negative, [A] [B] is generated by one element, which is an extension of the notion of lclm s. Indeed, any nonzero element with lowest degree in [A] and [B] is a generator. Proposition 5.3 The degree of a generator of [A, B] is (m + n d). Proof Since S d 0, Θ n d 1 A,..., A, Θ m d 1 B,..., B are F -linearly independent. It follows that a generator of [A] [B] is of degree greater than (m + n d 1). So, the lowest degree of nonzero elements in [A] [B] is no less than (m + n d). We consider two cases. First, assume that deg S d 0. In this case S d is regular. Thus, deg S d = d. Since S d 1 = 0, expanding S d 1 by Remark?? yields s n d Θ n d A + + s 0 A + t m d Θ m d B + + B = 0, (9) where the s s and t s belongs to F and s n d is nonzero because S d is regular. Let L = s n d Θ n d A + + s 0 A. The degree of L is (m + n d). Since rem(l, B) = 0 by (??), L belongs to [A] [B]. Hence, [A] [B] = [L]. Second, assume that deg S d = 1. Expanding S d by Remark?? yields s n i Θ n i A + + s 0 A + t m i Θ m i B + + B = rx 1 (10) where the s s, t s and r belong to F, r is nonzero and s n i is nonzero, where d + 1 i n. Applying Θ to (??) yields u n i+1 Θ n i+1 A + + u 0 A + v m i+1 Θ m i+1 B + + B = wx 1 (11) where the u s, v s and w belong to F. In particular, u n i+1 is nonzero because it is equal to σ(s n i ) by (??). Cancelling the inhomogeneous components of (??) and (??), we find that ru n i+1 Θ n i+1 A + (ru n i ws n i )Θ n i A + + (ru 0 ws 0 )A +rv m i+1 Θ m i+1 B + (rv m i wt m i )Θ m i B + (rv 0 wt 0 )B = 0. As Θ n d 1 A,..., A, Θ m d 1 B,..., B are F -linearly independent, i must be equal to (d+1). Therefore, L = ru n d Θ n d + (ru n d 1 ws n d 1 )Θ n d 1 A + (ru 0 ws 0 )A is a generator of [A] [B]. A generator of [A] [B] is a proper extension of the notion of the lclm of two elements of an Ore polynomial ring. The equality where Ā, B F [X], is generalized to deg Ā + deg B = lclm(ā, B) + gcrd(ā, B), deg A + deg B = deg L + d

Determinants for Inhomogeneous Equations 119 where L is a generator of [A] [B]. The proof of Proposition?? points out how to express some generators of [A] [B] by determinants. At last, we describe a geometric meaning of isolated subresultant in differential case. Assume that F [X] F is given in Example??. For an element G F [X] F, the solution set of G in a differential closed field of F is denoted by V G. If deg G 0, then V G is a translation of a C-vector space with dimension deg G. If deg G = 1, then V G is empty. Hence, deg G has some geometric meaning. Corollary 5.4 Let L be a generator of [A] [B]. 1. S d is regular deg A + deg B = deg L + deg S d. 2. S d is isolated deg A + deg B > deg L + deg S d. Proof The first assertion is immediate from Proposition??. The second one follows from the fact that S d is isolated if and only if deg S d < d. References [1] Abbot, J., Bronstein, M., Mulder, T. (1999): Fast deterministic computation of determinants of dense matrices, in: Proceedings of ISSAC 99, Dooley, S., Ed., ACM Press, 197 204. [2] Bronstein, M., Petkovsek, M. (1996): An introduction to pseudo linear algebra, Theoretical Computer Science, 157, 3-33. [3] Brown, W.S. (1971): On Euclid s Algorithm and the Computation of Polynomial Greatest Common Divisors JACM 18, 478 504. [4] Brown, W.S., Traub, J.F. (1971) On Euclid s Algorithm and the Theory of Subresultants. JACM 18, 505 514, 1971. [5] Collins, G.E. (1971) The Calculation of Multivariate Polynomial Resultants, JACM 18, 515 532. [6] Collins, G.E. (1967): Subresultant and Reduced Polynomial Remainder Sequences, JACM 16, 708 712. [7] Li, Z. (1998): A subresultant theory for Ore polynomials with applications, in: Proceedings of ISSAC 98, Gloor, O. Ed., ACM Press, 124 131. [8] Li, Z. (1996): A Subresultant Theory for Ore Polynomials and its Applications. PhD Thesis, Research Institute for Symbolic Computation, Johannes Kepler University, Linz, A-4040, Austria. [9] Li, Z., Nemes, I. (1997): A modular algorithms for computing greatest common right divisors of Ore polynomials, in: Proceedings of ISSAC 97, Küchlin, W. Ed., ACM Press, 282 298. [10] Loos, R. (1982) Generalized Polynomial Remainder Sequence, in Buchberger, B. Collins, G.E., and Loos, R. (eds.), Computer Algebra, Symbolic and Algebraic Computation, [11] Mishra, B. (1993) Algorithmic Algebra. Texts and Monographs in Computer Science, D. Gries D and Theory of Non-commutative Polynomials, Annals of Math., 34, 480-508. [12] Ore, O. (1933): Theory of Non-commutative Polynomials, Annals of Math., 34, 480-508.