Consequences of the meromorphic equivalence of standard matrix differential equations

Publicacions Matem`atiques, Vol 41 (1997), 613–630. CONSEQUENCES OF THE MEROMORPHIC EQUIVALENCE OF STANDARD MATRIX DIFFERENTIAL EQUATIONS H. J. Zwiesler Abstract In this article we investigate the question how meromorphic differential equations can be simplified by meromorphic equivalence. In the case of equations of block size 1, which generalizes the case of distinct eigenvalues, we identify a class of equations which are simplest possible in the sense that they carry the smallest number of parameters whithin their equivalence classes. We also discuss conditions under which individual equations can be simplified. Particular attention is paid to the requirement that the involved transformations can be explicitly computed. 1. Introduction Meromorphic equivalence has always been an important tool for the discussion of systems of linear differential equations in the complex plane. Among the most prominent examples are the contiguous relations for the hypergeometric functions. In recent years W. B. Jurkat’s book ([Jur]) stimulated many research efforts for a systematic investigation of meromorphic equivalence. A main goal is the identification of representatives under this equivalence relation which should then be the central objects of further function —theoretic inquiries. For many purposes it is important that this treatment is as explicit as possible. In a previous article ([JZ1]) W. B. Jurkat and the author studied these questions for differential equations of arbitrary dimension and block size 1 (definitions Keywords. Meromorphic differential equations, meromorphic equivalence, reduction theory, standard equations, isoformal, isomonodromy, piecewise algebraic functions. 1991 Mathematics subject classifications: Primary: 34A20; Secondary: 34A05, 34A30. 614 H. J. Zwiesler can be found in section 2). They showed that each equation is equivalent to an equation of a particularly simple form, a so-called standard equation. These standard equations allow a thorough and explicit study of all possible transformations. It is a natural question in this context to ask: How can these standard equations be further simplified by means of meromorphic equivalence? The main goal of the present article is to show that standard equations are simplest possible in the sense that they carry the smallest number of parameters within their equivalence classes. They also allow a natural parametrization and are therefore particularly well suited for further function —theoretic studies, e.g. how connection coefficients depend on the parameters of an equation. In section 2 we define the standard equations and introduce the concept of direct transformations and equivalence. Direct equivalence is closely related to meromorphic equivalence but turned out to simplify the discussion considerably since it avoids the use of constant similarities. These constant similarities would introduce unnecessarily complicated normalizations of the differential equations (compare [JZ2, Proposition 7]). Section 3 contains the definition of piecewise algebraic functions and a summary of their central properties. These piecewise algebraic functions occur naturally in the explicit formulae for the direct transformations and the transformed differential equations (Proposition 1). They provide the basic key for section 4 which forms the central part of this paper. It is devoted to the discussion of the number of parameters of a standard equation (Lemma 2) and the fact that the totality of standard equations cannot be obtained from a subclass with fewer parameters (Theorem 1). We can even discuss whether individual equations allow simplifications (Theorem 2), e.g. whether an equivalent equation exists for which more parameters vanish. This discussion reveals new invariants for standard equations. All of these results also apply to natural subclasses of our standard equations, e.g. to the important cases of equations with fixed formal invariants (isoformal) resp. with fixed monodromy (isomonodromy). This transfer is carried out in section 5 (Corollary 1 and Remark 5). In this context we compute the number of parameters for isoformal equations in Lemma 3. Although the formulae for the transformations are in general made up from several pieces we prove in section 6 that for the vast majority of equations one system of rational formulae suffices (Theorem 3). Finally we show in section 7 that the previous results carry over from Meromorphic Equivalence of DE’s 615 direct to meromorphic equivalence after an additional normalization of the standard equations (Theorems 1’, 2’, 3’). The reader is assumed to be familiar with the basic facts of the general theory of the meromorphic differential equations which can be found in [Jur]. Besides that we rely on the results of the articles [JZ1] and [JZ2]. 2. Standard Equations and Direct Equivalence This section contains a discussion of the differential equations and their transformations which are the basis for our investigations. A detailed treatment can be found in [JZ1]. Ameromorphic differential equation is a linear system of the form X(z)=A(z)X(z) where A(z) can be expanded in a convergent Laurentseries in a neighborhood of infinity with finite singular part and X(z) denotes a fundamental solution matrix (all matrices have dimensions n×nwith n≥2). For abbreviation we will denote such a differential equation by A(z) as long as no confusion is possible. When we replace X(z)byT(z)Y(z) where Tis a meromorphic transformation (i.e. Tand T−1are meromorphic at infinity) then Y(z)isa fundamental solution matrix for the equation B=T−1AT −T−1T.We say that Aand Bare meromorphically equivalent. Definition. The differential equation X=AX is called a standard equation if (i) A(z)= r  k=0 Akzk−1with r∈N(Poincar´e rank) and Ar=0, (ii) it possesses a formal solution H(z) of the form F(z)zΛeQ(z)where Q(z) = diag(q1(z),... ,q n(z)) is a polynomial matrix without constant term, Λ= diag(λ 1,... ,λ  n) is a complex matrix (called formal monodromy) whose diagonal elements satisfy λ j≡ λ k(mod 1) whenever qj=qkfor j=k(we summarize the conditions for Qand Λby saying that the equation has block size one) and the formal series F(z) satisfies F(z)=I+ ∞  j=1 Fjz−j, (iii) the eigenvalues of A0which are not equal, are already incongruent (mod 1). Remark 1 ([JZ1, Remarks 1 and 2]).The formal solution (I+ ∞  j=1 Fjz−j)zΛeQ(z)is uniquely associated with Aand denoted by 616 H. J. Zwiesler HA=FAzΛ AeQA(the subscript .Astands for a quantity that is uniquely determined by A). The special form of F(z) implies that the Poincar´e rank of Ais minimal. Such a standard equation also possesses an actual solution E(z)zMwhere E(z) is entire with det E(z)=0(∀z∈C) and Mis a lower triangular Jordan canonical form of A0(called actual monodromy). It is uniquely determined if we prescribe the ordering of the Jordan blocks. For that purpose we choose all complex zwith 0≤Re(z)<1 as a system of representatives modulo 1 and require: (i) the eigenvalues of Mare arranged such that the corresponding representatives are non-increasing (where here and in the sequel complex numbers are ordered lexicographically with real part first), (ii) the sizes of consecutive Jordan blocks for the same eigenvalue do not increase. This unique monodromy Mwill be denoted by MA. Definition. The standard equations Aand Bare said to be directly equivalent, if a meromorphic transformation Texists satisfying HA=THB. Such a Tis called a direct transformation from Ato B. Remark 2 ([JZ1, Remark 6]).Two standard equations Aand Bare meromorphically equivalent if and only if a permutation of Ais directly equivalent to some B∗which is diagonally similar to B. The occuring permutation is uniquely determined by QA,Λ  A,QB,Λ  B. The direct transformation is the central part of every meromorphic transformation but has the advantage to avoid the possible constant matrices that can be applied to HBand create the necessity to normalize A(z) further. Therefore we will concentrate mainly on direct equivalence. 3. Piecewise Algebraic Functions The coefficients of a direct transformation depend algebraically on the coefficients of the differential equation to which it is applied. This behavior implies the properties that are essential for the present reduction theory. Hence we will now introduce the concept of piecewise algebraic functions, give some of their basic properties and describe how they arise in the context of direct transformations. Definition ([JZ2]).Let Xbe a non-empty subset of CN(N∈N). A set of the form {x∈X:g(x)=0,g j(x)=0forj=1,... ,m}with m∈N0,g,gj∈Q[x] is called a P-set in xrelative to X.APA-set Meromorphic Equivalence of DE’s 617 relative to Xconsists of the finite union of P-sets relative to X. Its (algebraic) dimension over F(for an arbitrary field Fwith Q⊆F⊆C)is by definition the maximal number of algebraically independent components for each of its elements (where the empty set has dimension −1). For F=Qwe simply use the term dimension. In the case of a PA-set F⊆X×CM(M∈N) we say that Fis a PA-relation relative to X. Amultivalued function f:CN−→CMassociates with every x∈CN the finite set f(x)⊆CM(which may be empty). Its domain is defined as Df={x∈CN:f(x)=∅} and we also introduce the image of a set A⊆CNas f(A)=x∈Af(x) and the preimage of a set B⊆CM as f−1(B)={x∈CN:f(x)∩B=∅}. Furthermore we say that F={(x, y):y∈f(x)}⊆CN×CMis the graph of f. If the graph F is a PA-relation relative to Xthen fis called a PA-function relative to X. Finally, a PR-function relative to Xis defined to be a PA-function relative to Xwhere each set f(x) contains at most one element. It is important to observe that fis a PR-function relative to Xif and only if there exists a decomposition of Xinto PA-sets S0,... ,S m (m∈N0)relative to Xsuch that fis defined on X¬S0(difference of sets) and the components of fallow rational representations on each Sj for 1≤j≤m(decompositions are always disjoint). A first important property of PA-functions is the fact that the composition of PA-functions f:X−→CMand g:Y−→CM—defined as g(f(x)) = y∈f(x)g(y)— with ∅=Y⊆CMand f(x)⊆Yfor all x∈X is again a PA-function relative to X. Before we can relate these definitions to our direct transformation we must explain which parameters we will use to represent our standard equations. For that purpose we fix the Poincar´e rank rof A(z) and combine the entries of the matrices Ar−1,... ,A 0and of the diagonal of Ar in some fixed way in order to obtain a parameter vector in Crn2+n. The totality of parameter vectors corresponding to standard equations forms the parameter space X. Sometimes it is useful to extend the parameter vectors by including the eigenvalues λ1,... ,λ nof the monodromy matrix in the same order as they occur on the diagonal of MA. This leads to the parameter space X⊆Crn2+2n, and to every standard equation there corresponds exactly one parameter vector in Xresp. in X. Proposition 1. Let diagonal matrices Kand Kwith integer entries and equal traces be given. To every standard equation Athere exists at most one directly equivalent standard equation Bwith MA=MB+K,QA=QB,Λ A=Λ  B+Kand the function which 618 H. J. Zwiesler maps the parameter vector of Aonto the parameter vector of Bis a PRfunction (Crn2+2n−−→Crn2+2n)relative to X. On the other hand, every standard equation Bwhich is directly equivalent to Acanbeobtained by an appropriate choice of Kand K. Proof: This is essentially Corollary 2 and a consequence of Proposition2in[JZ1] with the difference that we no longer include the entries of Qor Λinto the parameter vectors. Due to Proposition 6 of [JZ2]we know that Qand Λcan be computed from Aby means of PR-functions. Hence we can compose the two PR-functions to obtain a PR-function in the parameter vectors of X. Remark 3. In [JZ1, Corollary 2] and [JZ2, Theorem 9] we also gave an explicit procedure to compute these PR-functions, but this is of minor importance for our present investigations. Although the eigenvalues of MAcoincide with those of A0and hence depend algebraically on the parameters of A, the parameters of Bare not PA-functions relative to Xsince we had to demand the ordering of these eigenvalues in connection with the matrix K. But we can say that the parameter vectors in Xare a part of a PA-function in the parameter vector in X. The closer study of the implications of Proposition 1 relies on some properties of PA-functions. For their formulation we need the following Definition. For a set S⊆RN(N∈N) we define the p-dimensional outer Hausdorff measure (p∈R,p≥0) as Hp(S) = sup ε>0 inf ∞  j=1 dp(Aj) where S⊆ ∞  j=1 Ajand the diameters d(Aj)<εfor all j. Moreover Sis called σp-finite if S= ∞  j=1 Sjwith Hp(Sj)<∞for all j; if in addition Hp(S)>0 then we say that Shas precise dimension p. We also use these real concepts for our complex sets by splitting each variable into its real and imaginary part. Therefore, the dimensions are doubled when we compare them to what we would expect from counting the parameters. Lemma 1. Let fbe a PA-function (CN−→CM)relative to X.For given A⊆Xthe following hold (Fa field with Q⊆F⊆C,p∈R, p≥0): Meromorphic Equivalence of DE’s 619 (i) the algebraic dimension over Fof f(A)is not greater than the algebraic dimension over Fof A, (ii) Hp(A)=0⇒Hp(f(A)) = 0, (iii) Ais σp-finite ⇒f(A)is σp-finite. Proof: See [JZ2, Proposition 2]. From this lemma it follows in particular that, if fpossesses an inverse that is a PA-function (relative to f(X)), then fpreserves the precise dimension of a set. The existence of such an inverse is equivalent to the fact that f−1({y}) is a finite set (∀y∈CM) according to [JZ2, end of section 2]. 4. On the Number of Parameters After we have made the necessary preparations we can now explain that our standard equations cannot be reduced further by means of direct equivalence. For that purpose we start by counting the parameters. Lemma 2. The parameter spaces Xand Xfor the standard equations with fixed Poincar´e rank rhave precise dimension 2(rn2+n). Proof: The standard equations incorporate in particular those equations where Arhas distinct eigenvalues and A0has incongruent eigenvalues. Their parameter vectors in Xform an open, non-empty subset of Crn2+n. Therefore, H2(rn2+n)(X)>0 holds. Furthermore, Crn2+nis σ2(rn2+n)-finite which is then also true for every subset. If we associate with every parameter vector in Xthe vector itself and the eigenvalues of A0in any possible order, then we obtain a PA-function (Crn2+n−→Crn2+2n) relative to X([JZ2, Lemma 3]). The image of X under this function is σ2(rn2+n)-finite due to Lemma 1(iii), and this also holds for Xas a subset of this image. In addition, H2(rn2+n)(X)=0 is impossible since this would imply H2(rn2+n)(X) = 0 by Lemma 1(ii) as Xis the projection of Xand projections are PA-functions ([JZ2, Lemma 4(i)]). It is worthwhile to notice that Theorem 9 in [JZ2] shows that the set Xis a Borel-set and therefore measurable. Of course, the same holds for X. Remark 4. Let us consider a collection of standard equations whose parameter vectors constitute a set S⊆X; at the same time they yield a set S⊆Xwhen we include the eigenvalues of the monodromy matrix. 620 H. J. Zwiesler Then the proof of Lemma 2 shows that Sis the image of Sunder a PAfunction, that Sis a subset of the image of Sunder a PA-function and that Ssatisfies Hp(S) = 0 resp. is σp-finite resp. has precise dimension pif and only if Ssatisfies Hp(S) = 0 resp. is σp-finite resp. has precise dimension p. Moreover, both sets have the same algebraic dimension over any field Fwith Q⊆F⊆C. Now we can formulate precisely what we mean when we say that our standard equations cannot be reduced further. Theorem 1. Let S⊆Xsatisfy Hp(S)=0(resp. be σp-finite resp. have precise dimension pfor some p≥0) and denote by S∗⊆X the parameter vectors of all standard equations which are directly equivalent to those whose parameter vectors belong to S. Then Hp(S∗)=0 (resp. S∗is also σp-finite resp. has precise dimension p). Proof: We will prove the theorem in an equivalent version namely with Xreplaced by X(see Remark 4). According to Proposition 1 any standard equation Bwhich is directly equivalent to a given Aleads to two diagonal matrices K=MA−MBand K=Λ  A−Λ Bwith integer entries and equal traces. By the σ-subadditivity of Hpit is sufficient to consider a fixed choice of Kand K. But then at most one Bexists which is directly equivalent to A. We learn from Proposition 1 that the parameter vector of Bis a PR-function relative to Xof the parameter vector of A, and due to Lemma 1(ii) such a PR-function maps nullsets onto nullsets. The other parts of the claim follow in exactly the same way by Lemma 1(iii) and the remark following it. An immediate consequence of Theorem 1 is that we cannot obtain all standard equations by means of direct equivalence from a subcollection whose parameter vectors form a set Swith H2(rn2+n)(S) = 0. Therefore if the totality of all standard equations can be generated from Sthen S has precise dimension 2(rn2+n). It is in this sense that our standard equations allow no further reduction and are simplest possible. Of course, Theorem 1 also applies to natural subclasses of our standard equations and we will devote section 5 to the discussion of such a case. But it is also interesting to investigate individual equations and ask whether they can be simplified. For that purpose we must first explain what the term “simplified” should mean. One natural interpretation is that we use the transcendency degree of the parameters of the equation and consider an equation to be simpler than another if its parameters have a lower transcendency degree. Theorem 2. If the standard equations Aand Bare directly equivalent Meromorphic Equivalence of DE’s 621 then their parameters have equal transcendency degrees over F(where F is any field with Q⊆F⊆C). Proof: Again we can use Remark 4 to find out that it makes absolutely no difference in the statement whether we use the parameters from Xor those from X. Thus we consider the parameters from Xand notice that the parameters of Bare PR-functions in those of A. Hence Lemma 1(i) applies and the transcendency degree over Fof the parameters of B cannot by greater than that of the parameters of A. But since the situation is symmetric in Aand Bthe two transcendency degrees must in fact be equal. This theorem leads to the interesting observation that for every field F(Q⊆F⊆C) we obtain an invariant under direct equivalence, namely the transcendency degree over Fof the parameters of the standard equation. One consequence of Theorem 2 is the fact that standard equations, all of whose parameters (from X) are algebraically independent over Q, can never be directly transformed into standard equations where one or more parameters vanish. 5. Isoformal Equations In this section we will apply Theorem 1 to those subclasses of standard equations which are obtained by fixing the formal invariants. Every standard equation Ais formally and directly transformed by FA to Q A(z)+Λ  Az−1. The only direct transformations connecting such diagonal standard equations are zKwith arbitrary, diagonal Kpossessing integer entries. Therefore we should introduce the diagonal matrix Λ∗ A whose eigenvalues have real parts in [0,1) such that Λ A−Λ∗ Ahas integer entries. Then QA(z) and Λ∗ Aform a complete system of formal direct invariants for standard equations. They are easily computable as can be seen from Theorem 9 in [JZ2]. The fact that they remain unchanged under direct transformations suggests that we should concentrate our attention to sets of equations for which these invariants agree. Such equations are called isoformal. This is an equivalence relation which yields equivalence classes of isoformal equations in the parameter spaces X. They are the objects of our further studies. First we aim at counting the parameters of such an equivalence class. For that purpose we define d(j, k) = max{deg(qj(z)−qk(z)),0}for 1 ≤j, k≤n. Here the degrees are defined as follows: deg(T∗) is the maximal occuring power of zin the Laurent expansion of T∗at infinity and deg0(T∗) 628 H. J. Zwiesler Lemma 2’. The parameter spaces Yand Yfor the normalized standard equations with fixed Poincar´e rank rhave precise dimension 2(rn2+1). Proof: Let us first consider those normalized equations for which all entries a(r−1) 1jin Ar−1(2 ≤j≤n) are 1. Then the other rn2+ 1 parameters in Aare only restricted by the requirement that Ais a standard equation. This is guaranteed if e.g. Arhas distinct eigenvalues and all eigenvalues of A−1are incongruent modulo 1. Hence the parameter vectors of these equations form a set of precise dimension 2(rn2+1)inY and hence also in Y. In the remaining cases at least one entry a(r−1) 1j(2 ≤j≤n) is zero and this forces another parameter to be 0 or 1. Therefore the set of the corresponding parameter vectors is σ2rn2-finite. Proposition 1’. Let a permutation matrix Pand diagonal matrices K,Kwith integer entries and equal traces be given. For any normalized standard equation Athere exists at most one meromorphically equivalent normalized standard equation Bwith MA=MB+K,PQAP−1=QB, PΛ AP−1=Λ  B+Kand the function which maps the parameter vector (from Y)ofAonto the one of Bis a PR-function relative to Y. Furthermore every normalized standard equation Bwhich is meromorphically equivalent to Ais obtained by an appropriate choice of P,K and K. Proof: This is a consequence of Proposition 2 and Corollary 2 in [JZ1] where the permutation can be applied to Ain a preliminary step. Again we use that the entries of the diagonal elements in QAand Λ Aare PR-functions relative to Y. With these informations we can now easily transfer Theorem 1, 2, and 3 to this situation. Theorem 1’. Let S⊆Ysatisfy Hp(S)>0(resp. be σp-finite resp. have precise dimension p) for some p≥0and denote by S∗⊆Y the parameter vectors of all normalized standard equations which are meromorphically equivalent to those whose parameter vectors belong to S. Then Hp(S∗)=0(resp. S∗is also σp-finite resp. has precise dimension p). In particular, if S∗is the totality of all normalized standard equations then Shas precise dimension 2(rn2+1). Theorem 2’. If the normalized standard equations Aand Bare meromorphically equivalent then their parameters have equal transcendency degrees over F(where Fis any field with Q⊆F⊆C). Meromorphic Equivalence of DE’s 629 Theorem 3’. Let P,K and Kbe given as in Proposition 1’. Then there exists a set S⊆Crn2+nand a vector fof rn2+2nrational functions from Q(y1,... ,y rn2+2n)with the following properties: (i) The set Y¬S⊆Crn2+nis σ2rn2 -finite. (ii) If the parameter vector a∈Yof the normalized standard equation Abelongs to Sthen b=f(a)—where ais the parameter vector of Ain Y— is the parameter vector of the normalized standard equation Bwhich is meromorphically equivalent to Aand satisfies MA=MB+K,PQAP−1=QB,PΛ AP−1=Λ  B+K. Proof: Here we take Sas the set of exactly those vectors from Crn2+n whose corresponding differential equations have a(r−1) nj = 1 for 1 ≤j≤ n−1 whereas all other rn2+1 parameters are algebraically independent over Q. Notice that such a vector is always the parameter vector of a normalized standard equation. Now we proceed along the same lines as in the proof of Theorem 3. The matrix E0is constructed in exactly the same way (notice that the normalizations do not restrict a(0) nn) and for r≥2 the remaining arguments remain valid since the normalization is carried out rationally and each new normalized equation must still contain rn2+ 1 algebraically independent parameters which prevents a(r−1) nj (1 ≤j≤n−1) from vanishing and therefore produces uniform rational formulae when we normalize them to 1. For r= 1 some elements in A0 are normalized which forces us to discuss whether the parts - t1,... ,- tnof the eigenvectors in E0still have algebraically independent components. For that purpose we consider the last row of ˜ A-a 1...1ann - tj 1=λjtj 1 which yields n−1 k=1 tjk +ann =λj(1 ≤j≤n). Hence the eigenvalues depend algebraically on the n2−n+ 1 parameters tjk and ann, and the same is therefore true for the elements of A0 (= E−1 0MAE0). But since A0contains n2−n+ 1 algebraically independent elements, the tjk must be algebraically independent, too. All other arguments carry immediately over to this situation. References [Jur] W. B. Jurkat,“Meromorphe Differentialgleichungen,” Lecture Notes in Mathematics 637, Springer, Berlin, 1978. 630 H. J. Zwiesler [JZ1] W. B. Jurkat and H. J. Zwiesler, On the Meromorphic Equivalence of Standard Matrix Differential Equations, Asymptotic Anal. 1(1988), 303–316. [JZ2] W. B. Jurkat and H. J. Zwiesler, Piecewise Algebraic Functions, Adv. Appl. Math. 11 (1990), 247–281. [Was] W. Wasow,“Asymptotic Expansions for Ordinary Differential Equations,” John Wiley & Sons, New York, 1965. Abteilung Unternehmensplanung Ulm University D-89069 Ulm GERMANY e-mail: [email protected] Primera versi´o rebuda el 4 de Setembre de 1996, darrera versi´o rebuda el 2 de Setembre de 1997