Formal Solutions of Singularly Peturbed Linear Differential Systems
FFormal Reduction of Singularly-Perturbed Linear Di ff erentialSystems Moulay A. Barkatou
XLIM UMR 7252 ; DMIUniversity of Limoges; CNRS123, Avenue Albert Thomas87060 Limoges, [email protected]
Suzy S. Maddah ∗ INRIA Saclay ˆ Ile-de-FranceB ˆ atiment Alan Turing1 rue Honor d’Estienne d’Orves91120 Palaiseau, [email protected] Abstract
In this article, we discuss formal invariants of singularly-perturbed linear di ff erential systems inneighborhood of turning points and give algorithms which allow their computation. The algo-rithms proposed are implemented in the computer algebra system M aple . Keywords:
Singularly-perturbed linear di ff erential systems, turning points, cyclic vectors, rankreduction, singularities, exponential part, formal solutions, computer algebra.Let ∂ = ddx and consider the decisively simple-looking di ff erential equation [34, Introduction,Subsection 1.5 ] ε ∂ f − ( x − ε ) f = x , ε )-region D : | x | ≤ α , < | ε | ≤ ε for some positive constants α and ε . Setting F = (cid:34) f ε∂ f (cid:35) , this equation can be rewritten as the following first-order system: ε ∂ F = A ( x , ε ) F = (cid:34) x − ε (cid:35) F . (1)One can observe that the Jordan form of A ( x ) : = A ( x ,
0) is not stable in any neighborhood of x =
0. We refer by turning points to such points where the Jordan form of A ( x ) changes, i.e.,either the multiplicity of the eigenvalues or the degrees of the elementary divisors change in aneighborhood of such points [40, p. 57]. For example, x = turning point for (1).Our goal is to compute a formal solution in a neighborhood of x = ε → ∗ A part of this work was developed within the author’s Ph.D. thesis at the University of Limoges, DMI.
Preprint submitted to Journal for revision October 18, 2018 a r X i v : . [ m a t h . C A ] D ec igure 1: • In the region β | ε | / ≤ | x | ≤ α where β is a positive constant, the transformation F = T G where T = (cid:34) x / (cid:35) transforms system (1) into( x − ε ) x / ∂ G = { (cid:34) (cid:35) + ( x − ε ) (cid:34) − − x / (cid:35) } G , which can be rewritten, by a self-explanatory change of notation x − ε = ξ , as ξ x / ∂ G = { (cid:34) (cid:35) + ξ (cid:34) − − x / (cid:35) } G . A transformation G = T U where T = (cid:34)
12 12 −
12 12 (cid:35) + O ( ξ ) would then result in the block-diagonalized system ξ x / ∂ U = { (cid:34) − (cid:35) + O ( ξ ) } U . Consequently, the system splits into two first-order linear di ff erential scalar equations. Wecan then write down the following fundamental matrix of formal solutions for (1) in thisregion: F outer = T T exp ( (cid:34) − x / ε − + O ( x − / ) 00 x / ε − + O ( x − / ) (cid:35) ) . • In the region | x | ≤ β | ε | / , we shall perform a so-called stretching transformation. Namelywe set τ = x ε − / and ∂ τ = dd τ . Then, for all τ such as | τ | < ∞ , except possibly for2eighborhoods of the roots of τ − = , the transformation F = L G where L = (cid:34) ε / (cid:35) reduces system (1) to ε / ∂ τ G = (cid:34) τ − (cid:35) G . (2)And the transformation G = L U where L = + τ + O ( τ ) + τ + O ( τ ) − i + i ω + O ( τ ) i − i τ + O ( τ ) , reduces system (2) to ε / ∂ τ U = { (cid:34) − i + i τ + O ( τ ) 00 i − i τ + O ( τ ) (cid:35) + O ( ε / ) } U . A fundamental matrix of formal solution of system (1) is then given by: F inner = L L exp ( − i τ + ( i / τ + O ( τ ) ε / i τ − ( i / τ + O ( τ ) ε / ) . We call F outer (resp. F inner ) an outer (resp. inner) solution as it is obtained in the outer (resp.inner) region around x =
0. The corresponding di ff erential systems are sometimes referred toas outer and inner di ff erential systems as well. However, unlike the above particular example,intermediate regions might be encountered.In this article, we study the construction of formal solutions (the formal reduction), of sys-tems of the following form: ε h ∂ F = A ( x , ε ) F , (3)where the entries of the matrix A ( x , ε ) are formal power series in ε , whose coe ffi cients are formalpower series in x ; and h is an integer which we call the ε -rank of the system.In [ ? , p. 74], Iwano summarized as follows the problems needed to be resolved to obtainthe complete knowledge about the asymptotic behavior of the solutions of a singularly-perturbedlinear di ff erential system:(P1) Divide a domain [ D ] in ( x , ε ) -space into a finite number of sub-domains so that the solutionbehaves quite di ff erently as ε tends to zero in each of these sub-domains; (P2) Find out a complete set of asymptotic expressions of independent solutions in each of thesesub-domains ;(P3)
Determine the so-called connection formula; i.e. a relation connecting two di ff erent com-plete sets of the asymptotic expressions obtained in (2). The turning points other than τ =
0, namely the roots of this equation do not explicitly correspond to the originalturning point. They are referred to as secondary turning points [34].
3n order to resolve (P1) and (P2), Iwano proceeded by associating a convex polygon to first-order systems in analogy with the scalar case, after imposing a precise triangular structure on A ( x , ε ). (P3) remains generally unresolved and it is out of the scope of this article as well (see,e.g. [20, 22, 38] and references therein). The hypotheses on A ( x , ε ) were eventually relaxedfor special types of systems in a series of papers by Iwano, Sibuya, Wasow, Nakano, Nishimoto,and others (see references in [40]). A famous and prevalent method among scientists is the W kb method (see, e.g. [26]), which, roughly speaking, reduces the system at hand into one whoseasymptotic behavior is known in the literature. However, as pointed out in [39], in view of thegreat variety and complexity of systems given by (3), it is rarely expected that a system can bereduced into an already investigated simpler form.The methods proposed in the literature for the symbolic resolution of such systems are notyet fully algorithmic, exclude turning points (see [18] and references therein), treat systems ofdimension two only, rely on Arnold-Wasow form [3], and / or impose further restrictions on thestructure of the input matrix. Moreover, there exists no computer algebra package dedicatedto the symbolic resolution of neither system (3) nor the scalar n th -order scalar equation. Thewidely-used softwares M aple and M athematica content themselves to the computation of outerformal solutions for scalar equations, and so does [27].In this article, we attempt to resolve (P2) algorithmically, that is give an algorithm to constructa fundamental matrix of formal solutions of an input system. We also attempt to give an insightinto (P1) within some examples.As illustrated by the introductory example, the formal reduction of a singularly-perturbedsystem given by (3) leads inevitably to the consideration of more general systems. These consid-erations will be discussed in Section 1. In Section 2, we give necessary preliminaries includingresults on n th -order scalar equations, the base cases, and the well-known Splitting lemma whichsplits the system into subsystems of lower dimensions, whenever possible. We discuss our con-tribution in the remaining sections, as we deviate from the classical reduction to give algorithmswhich act on the system directly without resorting to an equivalent scalar equation or to theArnold form: First we refine in Sections 3.1 and 3.2 the algorithms which we proposed in [2] forthe resolution of turning points and reduction of the ε -rank h to its minimal integer value. Thenin Section 4, we associate an ε -Newton polygon to a given system and show its role in retrievingformal invariants. And finally, we make use in Section 5 of these notions introduced to proposean overall formal reduction algorithm.As we have already indicated, the literature on this problem is vast. In the hope of keepingthis article concise but self-contained, we restrict our presentation to constructive results whichcontribute directly to our process of formal reduction. The omitted proofs are collected in theappendix along with illustrating examples.In this paper, we content ourselves to the process of formal reduction. Any reference to theasymptotic interpretation of formal solutions will be dropped in the sequel. One may consult inthis direction the method of matched asymptotic expansions (see, e.g. [37, 38] and referencestherein) or composite asymptotic expansions (see, e.g. [20, 22] and references therein).In the sequel, we adopt C as the base field for the simplicity of the presentation. However, anyother commutative field F of characteristic zero ( Q ⊆ F ⊆ C ) can be considered instead. Thealgorithms presented herein require naturally algebraic extensions of the base field. However,such extensions can be restricted as described in [8, Section 7.2].Since we are leading a local investigation, we assume that the input system has at mostone singular point, otherwise the region of study can be shrunk. Moreover, we place this singularpoint at the origin, otherwise a translation in the independent variable can be performed ( x → / x ∞ ). Notations • C is the field of complex numbers, Q is the field of rational numbers, and Q − = { q ∈ Q | q ≤ } . • We denote the dimension of the systems under consideration by n . We use r for the alge-braic rank of specified matrices (leading matrix coe ffi cients) and h for the ε -rank. • We use upper case letters for algebraic structures, matrices, and vectors; the lower caseletters i , j , k , l for indices; and some lower case letters (greek and latin) locally in proofsand sections, such as u , v , σ , ρ , (cid:37) , etc... • We use x , t , τ as independent variables and ∂ to signify a derivation. Since x dominates thisarticle, we drop it in the derivation. We set ∂ = ddx and ∂ τ = dd τ . • We use F for the unknown n -dimensional column vector (or n × n -matrix); we use G , U , W , and Z in this context as well. For scalar equations, we use f and g . • I d × n (resp. I n ) stands for the identity matrix of dimensions d × n (resp. n × n ); and O d × n (resp. O n ) stands for the zero matrix of dimensions d × n (resp. n × n ). If confusion isunlikely to arise, we simply use 0 to denote O d × n or O n . • We say that M ∈ M n (R) whenever the matrix M is a square matrix of size n whose entrieslie in a ring R. • GL n (R) is the general linear group of degree n over R (the set of n × n invertible matricestogether with the operation of matrix multiplication). • C [[ x ]] is the ring of formal power series in x whose coe ffi cients lie in C ; and C (( x )) is itsfraction field. The usual valuation of an element g ( x ) of C (( x )), i.e. the order of g in x , isdenoted by val x ( g ), with val x (0) = + ∞ . The ring C [[ x ]] (resp. the field C (( x ))) is endowedwith a di ff erential ring (resp. field) structure by considering the derivation ∂ . • C [[ ε ]] is the ring of formal power series in ε whose coe ffi cients lie in C ; and C (( ε )) isits fraction field. C [[ x ]][[ ε ]] (resp. C [[ x ]](( ε ))) is the ring of formal (resp. meromorphic)power series in ε whose coe ffi cients lie in C [[ x ]]. Similarly, C (( x ))[[ ε ]] (resp. C (( x ))(( ε )))is the ring (resp. field) of formal (resp. meromorphic) power series in ε whose coe ffi cientslie in C (( x )). • Any element f ∈ C (( x ))(( ε )) can be written as f = (cid:80) ∞ k ∈ Z f k ( x ) ε k , where f k ( x ) ∈ C (( x ))for all k ∈ Z . The derivation ∂ extends naturally to C (( x ))(( ε )) by the formula ∂ f = (cid:80) ∞ k ∈ Z ( ∂ f k ( x )) ε k . Moreover, the map val ε : C (( x ))(( ε )) → Z ∪ ∞ defines a valuation over C (( x ))(( ε )), satisfying the following properties for all f ( x , ε ) , g ( x , ε ) in C (( x ))(( ε )): – val ε ( f ) = ∞ if and only if f = – val ε ( f g ) = val ε ( f ) + val ε ( g ); – val ε ( f + g ) ≥ min ( val ε ( f ) , val ε ( g )).5 We give the blocks of a matrix M with upper indices, e.g. M = (cid:34) M M M M (cid:35) , and the size of the di ff erent blocks is dropped unless it is not clear from the context.
1. Ring of coe ffi cients In this first part, we investigate a suitable ring of coe ffi cients to treat first-order singularly-perturbed linear di ff erential systems and n th -order scalar singularly-perturbed di ff erential equa-tions, in a neighborhood of a turning point (see also [40, Chapter 2] and references therein). For( U , V ) ∈ R , we put P ( U , V ) = { ( U , V ) ∈ R | U ≥ U and V ≥ V } . Let f = (cid:80) ∞ k ∈ Z f k ( x ) ε k ∈ C (( x ))(( ε )) and let P f be the union of P ( k , val x ( f k )) , k ∈ Z . Then theNewton polygon of f , denoted by N f , is the boundary of the convex hull in R of the set P f . Wenow consider, for σ, p ∈ Q with σ ≤
0, the half-plane H σ, p = { ( U , V ) ∈ R | V ≥ σ U + p } . One can verify that K = { f ∈ C ((x))(( ε )) | N f ⊂ H σ, p for some σ ∈ Q − , p ∈ Q } is a di ff erential field endowed with the derivation ∂ = d / dx and it is the field of fractions of thering R = K ∩ C ((x))[[ ε ]]. Moreover, the elements of K can be characterized geometrically asfollows: f = (cid:88) k ∈ Z f k ( x ) ε k ∈ K ⇐⇒ ∃ σ ∈ Q − , p ∈ Q | ∀ k , val x (f k ) ≥ σ k + p . In fact, given a non-zero element f of K , we set ν f : = val ε ( f ). Among the half-planes whichcontain N f , we consider the half-planes which are bounded below by a straight line passingthrough the point ( ν f , val x ( f ν f )), i.e. the half-planes H σ, p f with p f : = val x ( f ν f ) − σν f . Then,among the H σ, p f ’s, we pick the half-plane of maximal slope, which we denote by σ f . Evidently, H σ f , p f is determined by a straight line which passes through the point ( ν f , val x ( f ν f )) and point(s)( k , val x ( f k )) for at least one k > ν . We denote by ( S f ) this straight line whose equation is givenby V = σ f U + p f , and we say that it is associated to f . For f =
0, we set σ f = p f = igure 2: Geometric interpretation of Example 1: ( S a ), ( S g ), ( S f ), ( S s ), are traced in black, green, blue, and redrespectively. Example 1.
Given a nonzero element f of C (( x ))(( ε )) . If we can construct a straight line offinite slope which passes through ( ν f , val x ( f ν f )) , and stays below the points ( k , val x ( f k )) for allk > ν , then f ∈ K . We illustrate the following examples in Figure 1: • Let a = x ε + x − ε = x ε (1 + x − ε ) then σ a = − . • Let g = (cid:80) ∞ k = x − k ε k then σ g = − . • Let f = x + (cid:80) ∞ k = x − k ε k then σ f = − . • Let s = x − + (cid:80) ∞ k = x − k ε k then σ s = − . The following two lemmas give further insight into K : Lemma 1.
If f ( x , ε ) ∈ K is non-zero, then there exists e ∈ Z − such that f ( x , x e ε ) ∈ C [[ x ]](( ε )) .Proof. Let f = (cid:80) ∞ k = ν f f k ( x ) ε k . We set µ k = val x ( f k ) and f k = x µ k ˜ f k . Then for any e ∈ Z − with e ≤ σ we have: f ( x , ε ) = ε ν f ( f ν f + f ν f + ε + f ν f + ε + f ν f + ε + . . . ) = ε ν f x µ ν f ( ˜ f ν f + x µ ν f + − µ ν f ˜ f ν f + ε + x µ ν f + − µ ν f ˜ f ν f + ε + x µ ν f + − µ ν f ˜ f ν f + ε + . . . ) = ε ν f x µ ν f ( ˜ f ν f + x µ ν f + − µ ν f − e ˜ f ν f + ( x e ε ) + x µ ν f + − µ ν f − e ˜ f ν f + ( x e ε ) + x µ ν f + − µ ν f − e ˜ f ν f + ( x e ε ) + . . . ) . But, µ k ≥ σ f ( k − ν f ) + µ ν f for all k ≥ ν f . Hence, µ k ≥ e ( k − ν f ) + µ ν f for all k ≥ ν f . We can thuschoose e to be the largest integer which is less than or equal to σ f . Lemma 2.
The field of constants of the di ff erential field ( K , ∂ ) is C (( ε )) .Proof. Let f ∈ K and suppose that ∂ f =
0. Then, (cid:80) ∞ k = ν f ( ∂ f k ( x )) ε k = ∂ f k ( x ) = k ≥ ν f . Thus, for every k ≥ ν f , f k ( x ) = c k for some c k ∈ C , which yields f = (cid:80) ∞ k = ν f c k ε k . 7enceforth, algorithmically, we can use the following handy representation of any element f ∈ K : f ( x , ε ) = ∞ (cid:88) k = ν f f k ( x ) ε k = ∞ (cid:88) k = ν f x − k σ f f k ( x )( x σ f ε ) k = x p f ∞ (cid:88) k = ν f x − k σ f − p f f k ( x ) ξ kf , with ξ f = x σ f ε. (4)Evidently, val x ( x − σ f k − p f f k ( x )) ≥ k ≥ ν f . Moreover, under the notation ξ f , we have: ∂ ( (cid:88) k = ν f f k ξ kf ) = (cid:88) k = ν f ( ∂ ( f k ) + σ f kx f k ) ξ kf . Motivated by the introductory example, and with the help of the above notations, we willinvestigate the following system rather than (3): ∂ F = A ( x , ε ) F where A ∈ M n ( K ) . (5)The valuation val ε (and the usual valuation val x of C (( x ))) extends naturally to matrices A = ( a i , j ) ∈ M n ( K ) by val ε ( A ) = min i , j val ε ( a i , j ). We can then define the Newton polygon of A ∈M n ( K ), ν A ∈ Z , p A ∈ Q , and σ A ∈ Q − , in an analogous manner to those of f ∈ K . Hence, wecan express A ( x , ε ) as follows: A = ∞ (cid:88) k = ν A A k ε kA , A ν A (cid:44) , p A = val x ( A ν A ) − σ A ν A , and val x ( A k ) ≥ σ A k + p A ∀ k ≥ ν A . (6)Hence by setting ξ A = x σ A ε , we can write in analogy with (4): A ( x , ε ) = x p A ∞ (cid:88) k = ν A x − k σ A − p A A k ( x ) ξ kA = ξ ν A A x p A ∞ (cid:88) k = x − k σ A − val x ( A ν A ) A ν A + k ( x ) ξ kA where x − val x ( A ν A ) A ν A | x = (cid:44) . Thus, we can rewrite system (5) as follows: ξ − ν A A x − p A ∂ F = ( ∞ (cid:88) k = x − k σ A − val x ( A ν A ) A ν A + k ( x ) ξ kA ) F , where x − k σ A − val x ( A ν A ) A ν A + k ∈ M n ( C [[ x ]]) . Similarly, if we consider a n th -order di ff erential equation whose coe ffi cients are elements of K : a n ( x , ε ) ∂ n f + a n − ( x , ε ) ∂ n − f + . . . + a ( x , ε ) ∂ f + a ( x , ε ) f = , (7)then setting F = [ f , ∂ f , ∂ f , . . . , ∂ n − f ] T , we can express (7) equivalently as a system ∂ F = A ( x , ε ) F where A ( x , ε ) ∈ M n ( K ) is a companion matrix, and we define ν a : = ν A , p a : = p A , σ a : = σ A , and ξ a : = ξ A . 8 .1.1. Notations Thus, in this paper, we treat system (5) in the following form:[ A σ A ] ξ hA x p A ∂ F = A ( x , ξ A ) F , A ( x , ξ A ) = ∞ (cid:88) k = A k ( x ) ξ kA ∈ M n ( R ) , A (x = (cid:44) , (8)where ξ A = x σ A ε , σ A ∈ Q − , p A ∈ Q , and h ∈ Z . Under the definition of ξ A , A ( x , ξ A ) = (cid:80) k = A k ( x ) ξ kA with A k ( x ) ∈ M n ( C [[ x ]]) for all k ≥
0. We refer to A ( x ) as the leading coe ffi cientmatrix and to A , : = A ( x =
0) as the leading constant matrix.We also consider n th -order scalar linear di ff erential equations of the form:[ a σ a ] ∂ n f + a n − ( x , ξ a ) ∂ n − f + . . . + a ( x , ξ a ) ∂ f + a ( x , ξ a ) f = , (9)where a i ( x , ξ a ) = (cid:80) ∞ k ∈ Z a i , k ( x ) ε k ∈ K for all i ∈ { , . . . , n } , with a i , k ( x ) ∈ C [[ x ]] for all k ≥ A σ A ] given by (8) can be also expressed as a scalar equation ofthe form (9). The theoretical possibility stems from the work of [19] which discusses cyclicvectors. Moreover, an algorithm to compute a companion block diagonal form of [ A σ A ] (andhence equivalent scalar form) can be obtained by generalizing the work in [7] developed forunperturbed systems (see Appendix A).In the sequel, for the clarity of the presentation, the index A (resp. a ) will be dropped from ν A , p A , σ A , and ξ A (resp. ν a , p a , σ a , and ξ a ) whenever confusion is unlikely to arise. Consider system [ A σ A ] given by (8). Let T ∈ GL n ( K ) then the transformation F = TG (alsocalled gauge transformation ) yields a system[ ˜ A σ ˜ A ] ξ ˜ A ˜ h x ˜ p ∂ G = ˜ A ( x , ξ ˜ A ) G , where ˜ A ∈ M n ( R ) . (10)for some σ ˜ A ∈ Q − , ˜ p ∈ Q , and ˜ h ∈ Z . We say that systems [ A σ A ] and T [ A σ A ] : = [ ˜ A σ ˜ A ]are equivalent . Examples of such transformations and their applications are the transformations T , T , L , L applied within the introductory example.In the sequel, we seek at each step a transformation which yields for an input system [ A σ A ],an equivalent system [ ˜ A σ ˜ A ], so that either ˜ h < h or [ ˜ A σ ˜ A ] can be decoupled into systems of lowerdimensions. Using a recursive process which employs this approach, we aim to break down anyinput system [ A σ A ] into system(s) which can be treated with known methods (see basic cases ofSection 2.1).Evidently, σ ˜ A might di ff er from σ A and hence ξ A will be updated to ξ ˜ A after the applicationof T (see the introductory example), which brings us to the next subsection. (P1)As we are lead naturally to the study of system [ A σ A ] with σ A ∈ Q − within the study ofsystem (8), we will need to understand the inevitable growth of the order of the poles in x withinthe reduction. For instance, in the introductory example, we start with an input system for which σ A = σ U = − k . From here initially stems our main motivation for our choice of the ring of coe ffi cientsand the representation in terms of ξ A : the orders of the poles in x which might be introduced in9n input system within the process of reduction, are stored in p and σ , which allows investigatingthem at any step within the reduction process. We can thus talk about a restraining index ρ = − /σ . This is the first step towards determining the inner and intermediate regions. In fact, thecomplicated behavior anticipated in the neighborhood of turning points can be investigated withthe help of a sequence of positive rational numbers:[ ρ ] 0 = ρ < ρ < ρ < · · · < ρ m . (11)With this sequence, the domain | x | ≤ x of [ D ] can be divided into a finite number of sub-domainsin each of which the solution of behaves quite di ff erently (see [23, Intro, pp 2] and [ ? , Chapter1]). In case [ ρ ] is known, one can apply adequate stretching transformations to the originalinput system, i.e. a change of the independent variable of the form τ = x ε − ρ i , i ∈ { , . . . , m } .Then, using our proposed algorithm, we can construct inner and intermediate solutions. In thispaper, we do not investigate (P1), i.e. we do not compute [ ρ ]. However, we motivate a plausibleapproach in some examples (see Appendix E).
2. Preliminaries
A detailed discussion of the scalar case ( n =
1) and regularly-perturbed systems ( h ≤ σ A = p A =
0. In this section, we give a briefdiscussion which discards this restriction on σ A and p A . ≤ h ≤ A σ A ] can be sought, up to any order µ , upon presentingthe solution as a power series in ξ A = x σ A ε , i.e. F = ∞ (cid:88) k = F k ( x ) ξ − h + kA . The latter can then be inserted into [ A σ A ] ξ h x p ∂ F = AF and the like powers of ξ A equated. Thisreduces the symbolic resolution to solving successively a set of inhomogeneous linear singulardi ff erential systems in x solely: x p ∂ F = ( A ( x ) + h σ A x p − I ) F x p ∂ F = ( A ( x ) + ( h − σ A x p − I ) F + A ( x ) F ... x p ∂ F µ = ( A ( x ) + ( h − µ ) σ A x p − I ) F µ + A ( x ) F µ − + . . . + A µ ( x ) F . For any k ≥
0, the solution of the homogeneous system singular in x can be sought via theM aple package I solde [14] or the M athemagix package L indalg [29] (which are both basedon [8]). Afterwards, the solution of the inhomogeneous system can be obtained by the methodof variation of constants (see, e.g. [5, Theorem 3, p. 11]). We remark that a transformation F k = x − k σ A G reduces the inhomogeneous system in the dependent vector F k to: x p ∂ G = ( A ( x ) + h σ A x p − I n ) G ( x ) , F ( x ). Thus we have: F k ( x ) = x − k σ A F ( x ) (cid:90) F − ( x ) x k σ A [ A ( x ) F k − ( x ) + . . . + A k ( x ) F ( x )] dx . The n th -order singularly-perturbed scalar di ff erential equations are treated in [27] upon in-troducing an analog to the Newton polygon and polynomials (see e.g. [6] and references thereinfor the unperturbed counterparts of these equations): the ε -polygon and ε -polynomials. In thissubsection, we adapt this treatment to the more general equation [ a σ a ] given by (9):[ a σ a ] ∂ n f + a n − ( x , ξ a ) ∂ n − f + . . . + a ( x , ξ a ) ∂ f + a ( x , ξ a ) f = . We first give an analog to [27, Definition 2.1] and then generalize [27, Proposition 4.1],whose proof is easily adaptable from [27] (see Appendix B).
Definition 1.
Consider the scalar equation [ a σ a ] given by (9) . For ( u , v ) ∈ R , we putP ( U , V ) = { ( U , V ) ∈ R | U ≤ U and V ≥ V } . Set ν i = val ε ( a i ) for i ∈ { , . . . , n } and let P a be the union of P ( i , ν i ) . Then the ε -polygon of [ a σ a ] ,denoted by N ε ( a ) , is the intersection of R + × R with the convex hull in R of the set P a . We denotethe slopes of the edges of N ε ( a ) by { e , . . . , e (cid:96) } . These slopes are non-negative rational numbers.And, for every j ∈ { , . . . , (cid:96) } , consider the algebraic equation given by ( E j ) (cid:96) (cid:88) k = a i k , val ε ( a ik ) ( x ) X ( i k − i ) = , where ≤ i < i < · · · < i (cid:96) = n denote the integers i for which ( i , ν i ) lie on the edge of slope e j of the ε -polygon, and a i ,ν i ( x ) = ξ − ν i a a i ( x , ξ a ) | ξ a = . We say that ( E j ) is the ε -polynomial associatedto the slope e j . Proposition 1.
Consider a nonzero [ a σ a ] given by (9) and its ε -polygon N ε ( a ) of slopes { e , . . . , e (cid:96) } .Let f ( x , ξ a ) = exp ( (cid:90) q ( x , ξ a ) dx ) with q ( x , ξ a ) (cid:44) ∈ (cid:91) s ∈ N ∗ C (( x ))(( ξ / sa )) . If f ( x , ξ a ) satisfies [ a σ a ] formally thenq ( x , ξ a ) = ξ e j a ( X ( x ) + O ( ξ a )) for some j ∈ { , . . . , (cid:96) } , and X ( x ) is one of the non zero roots of the ε -polynomial ( E j ) associatedto e j . The full expansion of q ( x , ξ a ) can be obtained with successive substitutions of the form f = exp ( (cid:82) X ( x ) ξ eja dx ) g [27, Proposition 4.2]. 11 igure 3: The ε -polygon associated to (12) in Example 2 Example 2.
Consider the following linear di ff erential equation with σ a = : [ a σ a ] ∂ f − x ξ a ∂ f − ξ a f = . (12) We wish to study this equation according to its ε -polygon (see Figure ). We have two slopes: • The slope e = for which ( E ) x X + = whose nonzero solutions are X = ± i √ x . • The slope e = for which ( E ) X − xX = whose nonzero solution is X = x.Thus, the leading terms (of the exponential part) of solutions of (12) are given by exp ( (cid:82) x ξ a dx ) = exp ( (cid:82) x ε dx ) and exp ( ± (cid:82) i √ x ξ / a dx ) = exp ( ± (cid:82) i √ x ε / dx ) . One can also refer to [27, Section 5] for examples on the harmonic oscillator with dampingand Orr-Sommerfeld equation.Based on the above, we give the following definition:
Definition 2.
Consider the scalar equation [ a σ a ] given by (9) . We call the largest slope of the ε -polygon N ε ( a ) , the ε -formal exponential order ( ε -exponential order, in short) : ω ε ( a ) = max { e j , ≤ j ≤ (cid:96) } . Clearly, under the notations of this subsection, we have: ω ε ( a ) = n − max i = (0 , − val ε ( a i ) n − i ) . Due to their equivalence, the formal solutions of a first-order system [ A σ A ] given by (8) canbe computed from an equivalent scalar equation [ a σ a ] given by (9), using the ε -polygon of [ a σ a ].12oreover, (11) can be computed as well using the Iwano-Sibuya’s polygon (which collects valu-ations with respect to both x and ε [23]). However such a treatment is unsatisfactory (see e.g. [27,Conclusion]) since it overlooks the information that we can derive from the system directly anddemands an indirect treatment. Nevertheless, it plays a key role in the theoretical basis of thealgorithm which we will develop in later parts. Most importantly, the equivalence between [ A σ A ]and [ a σ a ], and the invariance of the ε -exponential order under gauge transformations, allows thedefinition of the ε -exponential order of system [ A σ A ] as follows: Definition 3.
Consider a system [ A σ A ] given by (8) and an equivalent scalar equation [ a σ a ] givenby (9) . The ε -polynomials and ε -formal exponential order ω ε ( A ) of [ A σ A ] are those of [ a σ a ] . It follows that, given a system [ A σ A ], we have: ω ε ( A ) = max ≤ i ≤ n (0 , − val ε ( a i ) n − i ) . (13)where [ a σ a ] is a scalar equation equivalent to [ A σ A ].In this paper, we give a direct treatment of the system which will lead eventually to the com-putation of ω ε ( A ), the ε -polynomials, and consequently a basis of the space of formal solutions,without resorting to the equivalence of an input system to a scalar equation. Our treatment relieson the properties of the eigenvalues of the leading coe ffi cient of the input system. We distinguishbetween three cases (distinct, unique, or zero eigenvalues), the first of which is classical and isrecalled in the next subsection. ε -Block Diagonalization Consider a system [ A σ A ] given by (8). A classical tool in the perturbation theory is theso-called splitting which separates o ff the existing distinct coalescence patterns. Whenever theleading constant matrix A , : = A ( x =
0) admits at least two distinct eigenvalues, the systemcan be decoupled into subsystems of lower dimensions:
Theorem 1.
Consider system [ A σ A ] given by (8)[ A σ A ] ξ hA x p ∂ F = A ( x , ξ A ) F , A ( x , ξ A ) ∈ M n ( R ) , A , (cid:44) . If A , = (cid:34) A , OO A , (cid:35) such that A , and A , have no eigenvalues in common, then there exists aunique transformation T ( x , ξ T ) = (cid:80) ∞ k = T k ( x ) ξ kT ∈ GL n ( R ) , given byT ( x , ξ T ) = I + ∞ (cid:88) k = (cid:34) O T k ( x ) T k ( x ) O (cid:35) ξ T k , such that the transformation F = TG gives [ ˜ A σ ˜ A ] ξ h ˜ A x p ∂ G = ˜ A ( x , ξ ˜ A ) G = (cid:34) ˜ A ( x , ξ ˜ A ) OO ˜ A ( x , ξ ˜ A ) (cid:35) G where ˜ A , = A , . Moreover σ T , σ ˜ A ≥ σ where σ = σ A + p − h if p < σ A otherwise . A ( x ), followed by successiveblock-diagonalization of coe ffi cients (see, e.g., [21, Theorem XII-4-1, p. 381] and [18, Chapter3, p. 56-59] for the particular case of σ A = p A =
0, and [40, Theorem 8.1, p. 70-81] for thegeneral case). We give a constructive proof in Appendix B. We remark that the specified formof A , in Theorem 1 is non-restrictive. It su ffi ces that A , has at least two distinct eigenvalues.Such a block-diagonal form can be then attained by Jordan constant transformation. In case thebase field is not algebraically closed, the weaker transformation given in [15, Lemma A.1] canbe applied instead, since it does not require any algebraic field extensions.After splitting the input system, we proceed in our reduction for each of the decoupled sub-systems in parallel. We can thus assume, without any loss of generality, that the leading constantmatrix of the input system [ A σ A ] has a unique eigenvalue γ ∈ C . Thus, upon applying the so-called eigenvalue shifting, i.e. F = EG = G exp( (cid:90) ξ − hA x − p γ dx ) , (14)one can verify that the resulting system E [ A σ A ] has a nilpotent leading constant matrix. Thus, itremains to discuss this case of nilpotency. This is the goal of the two following sections.
3. Two-fold rank reduction
Without loss of generality, we can now assume that system [ A σ A ] given by (8) is such that A , is nilpotent. At this stage of reduction, our approach diverges from the classical indirectones which require Arnold-Wasow form, a cyclic vector, or reduction to a companion form. Twocases arise: • A , is nilpotent but A ( x ) is not: In this case, the eigenvalues of A ( x ) might coalesce insome neighborhood of x =
0, which would cause the so-called turning point. In Subsec-tion 3.1, we propose an algorithm to compute a transformation which reduces the inputsystem to a system for which both A , and A ( x ) are nilpotent. • Both A , and A ( x ) are nilpotent: In this case, we propose in Subsection 3.2 an algorithmto compute a transformation which can reduce the ε -rank h of system [ A σ A ] to its minimalnonzero integer value. This minimality is an intrinsic property of the system and its so-lutions. Its computation paves the way for the retrieval of the ε -exponential order of thesystem, as we show later in Section 4. We illustrate the source of turning points in the following example:
Example 3. [38, p. 223] Let a ( x ) , b ( x ) be holomorphic for x in a region Ω ∈ C and considerM ( x ) = x a ( x )0 x b ( x )0 0 0 . he transition matrix T ( x ) = α ( x ) β ( x ) γ ( x )( − a ( x ) x − + b ( x ) x − )0 α ( x ) − γ ( x ) b ( x ) x − γ ( x ) , with arbitrary scalarfunctions α ( x ) , β ( x ) , γ ( x ) such that α ( x ) (cid:44) and γ ( x ) (cid:44) for all x ∈ Ω , yields for x (cid:44) :J ( x ) = T − M ( x ) T ( x ) = x x
00 0 0 . If α ( x ) , β ( x ) , γ ( x ) are holomorphic, so is T , provided that zero is not a point of Ω . If ∈ Ω , thenT ( x ) has a pole at x = unless the following conditions are satisfied:b (0) = and ∂ b (0) = a (0) . Therefore, M ( x ) is in general not holomorphically similar to J ( x ) in regions that contain x = .Moreover, if the second condition is not satisfied, two possibilities arise: • if b (0) = then there exists a constant invertible matrix S such that S − M (0) S = . Then M ( x ) is pointwise similar to J ( x ) , even in regions containing x = . • if b (0) (cid:44) then the Jordan form of M (0) is J (0) = so that J ( x ) is not holomor-phic at x = . Hence, it might occur that the Jordan matrix is not holomorphic in some region or the Jordanmatrix is itself holomorphic but, nevertheless, not holomorphically (although perhaps point-wise)similar to M ( x ). Below is an example which exhibits a system whose leading coe ffi cient matrixhas such complications: Example 4. [40, p. 57] Consider the following system with σ A = , i.e. ξ A = ε : [ A σ A ] ξ A ∂ F = ξ A x F where A ( x ) = x . It follows from the form of A ( x ) that the origin is possibly a turning point. We remark that thissystem is equivalent to (12) , as it is obtained from the latter by setting F = [ f , ξ A ∂ f , ξ A ∂ f ] T . We recall the following proposition which we first gave in [2], and we refine its proof:
Proposition 2. [2, Section 3] Consider system (8) given by [ A σ A ] ξ hA x p ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ kA Fwith ξ A = x σ A ε . Suppose that the leading constant matrix A , is nilpotent and A ( x ) has at leastone nonzero eigenvalue. Then there exists an invertible polynomial transformation T in a root ofx such that the transformation F = TG and a re-adjustment of the independent variable x resultsin a system whose leading constant matrix has at least one nonzero constant eigenvalue. xample 5. Consider the matricial form of Weber’s equation ( σ A = , p A = ) given by: [ A σ A ] ξ A dFdx = (cid:34) x (cid:35) F , with ξ A = x σ A ε . Let T ( x ) = (cid:34) x (cid:35) then F = TG yields: [ B σ A ] ξ dGdx = (cid:34) xx − ξ A x − (cid:35) G . Upon factorizing x, we have: ξ A x − dGdx = (cid:34) − ξ A x − (cid:35) G . We compute σ ˜ A = − and thus set ξ ˜ A = x − ε . The above system can be rewritten as: [ ˜ A σ ˜ A ] ξ ˜ A x dGdx = (cid:34) − ξ ˜ A (cid:35) G . Clearly, ˜ A , has constant eigenvalues and so the splitting lemma can be applied to decouplethis system into two scalar equations.Proof. (Proposition 2) The eigenvalues of A ( x ) admit a formal expansion in the fractional pow-ers of x in the neighborhood of x = µ ( x ) = (cid:80) ∞ j = µ j x j / s be a nonzero eigenvalue of A ( x ) with s ∈ N ∗ , j ∧ s =
1, and whoseleading exponent, i.e. smallest j / s for which µ j (cid:44)
0, is minimal among the other nonzero eigen-values. Without loss of generality, we can assume that s =
1, otherwise we set x = t s . By [9, 24],there exists T ∈ GL n ( C ( x )) such that for B ( x ) = T − A ( x ) T = B , + B , x + . . . : θ B ( x ) ( λ ) = x rank ( B , ) det( λ I + B , x + B , ) | x = does not vanish identically in λ . Let ν > B ( x ) in x . By [24, Proposition1], there are n − deg( θ ) eigenvalues of B ( x ) whose leading exponents lie in [ ν, ν + θ )eigenvalues for which the leading exponent is equal to or greater than ν +
1. Then, applying thetransformation F = T ( x ) G to the system [ A σ A ] yields[ B σ A ] ξ hA x p ∂ G = B ( x , ξ A ) G = ∞ (cid:88) k = B k ( x ) ξ kA G where B ( x ) : = x ν ( B , + B , x + . . . ) and B , has n − deg( θ ) eigenvalues whose leading exponentslie in [0 , A ( x , ξ A ) = x − ν B ( x , ξ A ) then the order of poles introduced in x in ˜ A ( x , ξ A ) is atworst (span x ( T ) + ν ), where span x ( T ) is the di ff erence between the valuation and the degree of thepolynomial transformation T ( x ) in x . Hence, σ ˜ A ≥ σ A − span x ( T ) − ν and [ B σ A ] can be rewrittenas [ ˜ A σ ˜ A ] ξ h ˜ A x ˜ p ∂ G = ˜ A ( x , ξ ˜ A ) G = ∞ (cid:88) k = ˜ A k ( x ) ξ k ˜ A G ξ ˜ A = x σ ˜ A ε , ˜ p ∈ Z , ˜ A k ( x ) ∈ M n ([[ x ]]), ˜ A ( x ) = x − ν B ( x ), and ˜ A , = B , . Then µ ( x ) is aneigenvalue of ˜ A ( x ) with a minimal leading exponent and hence it is among those whose leadingexponents lie in [0 , s =
1, and hence the leading exponent of µ ( x ) is zeroand µ (cid:44)
0. Since µ is a nonzero eigenvalue of ˜ A , , it follows that the ˜ A , is non-nilpotent.We remark that the eigenvalues of A ( x ) are the roots of the algebraic scalar equation f ( x , λ ) = det( A ( x ) − λ I n ) = T can be computed via I solde , mini I solde , or L indalg . Remark 1.
Proposition 2 leads to the following observation about the detection of a turningpoint: Consider an input system [ A σ A ] given by (3) with σ A = , p A = , and restraining index ρ A = − /σ A . If [ A σ A ] has a turning point at x = then, by the end of the process of formalreduction of this system (i.e. whenever all decoupled subsystems are either of non-positive ε -rank or are scalar equations), at least one of its decoupled subsystems has a finite restrainingindex. Example 6.
Consider the following system whose σ A = and ξ A = ε : [ A σ A ] ξ A ∂ F = ξ A x F where A ( x ) = x . We first compute s = . We set x = t and compute T = t
00 0 t . Or equivalently, we considerT = x /
00 0 x . Then F = TG yields: ξ A ∂ G = x / { + x − / ξ A + − − x − ξ A } G , which we rewrite as: [ ˜ A σ ˜ A ] ξ A x / ∂ G = { + ξ ˜ A + − x
00 0 − x ξ A } G , where ξ ˜ A = x − / ε . Now that the leading term is a constant matrix with three distinct eigenvalues,we can proceed by applying the Splitting of Theorem 1. In the above example, the system could be decoupled thoroughly into three scalar equations.However, we might encounter di ff erent scenarios as well, as illustrated in the following example. Example 7. [40, p. 57] We recall the system of Example 9 (which is equivalent to (12) ) with ξ A = ε : [ A σ A ] ξ A ∂ F = ξ A x F . s mentioned before, we have a turning point at x = . A , is nilpotent and the eigenvalues ofA ( x ) are and x, whence s = . Let T = diag(1 , x , x ) then F = TG yields: ξ A ∂ G = x { + x − ξ A + − − x − ξ A } G . Setting ξ ˜ A = x − ε , the former can be rewritten equivalently as: [ ˜ A σ ˜ A ] ξ A x ∂ G = { + ξ ˜ A + − x
00 0 − x ξ A } G . The leading constant matrix ˜ A , is no longer nilpotent. Hence the system can be decoupled intotwo subsystems upon setting G = T W whereT = + − − − − − − − ξ ˜ A + O ( ξ A ) . The resulting equivalent system then consists of the two decoupled lower dimension systemswhere W = [ W , W ] T , σ B = σ C = σ ˜ A = − : [ B σ B ] ξ B x ∂ W = { (cid:34) (cid:35) + (cid:34) − − (cid:35) ξ B + (cid:34) − − + x (cid:35) ξ B + O ( ξ B ) } W . [ C σ C ] ξ C x ∂ W = { + ξ C + (1 + x ) ξ C + O ( ξ C ) } W . The exponential part (and hence formal solution) of the second subsystem is clearly (cid:82) ξ − C x − (1 + O ( ξ C )) dx = ε − x (1 + O ( ε − x )) . One can observe that the eigenvalue x of the leading matrixcoe ffi cient of the input system [ A σ A ] is recovered as expected. This is in accordance with theexponential parts obtained for (12) . As for the first subsystem, B ( x ) and B , are simultaneouslynilpotent. Due to this dual nilpotency, in order to proceed in the formal reduction of [ B σ B ] , wewill make use of the ε -rank reduction in the following subsection.3.2. ε -Rank reduction We consider again system [ A σ A ] given by (8). We assume without any loss of generality that A ( x ) and A , are simultaneously nilpotent. In this section, we investigate the ε -rank reductionof the system, i.e. we seek to determine the minimal integer ε -rank among all systems equivalentto [ A σ A ]. If this minimal integer rank turns out to be non-positive then we continue the reductionby treating the system as in Section 2.1.1. Otherwise, the minimal integer rank gives an upperbound for the ε -exponential order, which allows us to proceed to Section 4.In analogy to its unperturbed counterpart, we define the ε -Moser rank and ε -Moser invariantof system [ A σ A ] to be the following rational numbers respectively: m ε ( A ) = max (0 , h + rank ( A ( x )) n ) and µ ε ( A ) = min { m ε ( T [ A σ A ]) for all T in GL n ( K ) } . efinition 4. System [ A σ A ] (the matrix A ( x , ξ A ) respectively) is called ε -reducible if m ε ( A ) > µ ε ( A ) .Otherwise it is said to be ε -irreducible. If m ε ( A ) ≤
1, then h =
0. Hence, we restrict our attention to the case of m ε ( A ) > A ]; nor with the sense of reduced system already employed in the literature, i.e.the system whose matrix is the leading coe ffi cient matrix of [ A σ A ]: x p ∂ F = A ( x ) F . In fact, themotivation behind Moser’s work in [32] was to determine the nature of the singularity (regularor irregular) of an unperturbed system x p ∂ F = A ( x ) F , A ( x ) ∈ M n ( C [[ x ]]). Consequently, thereduction of the so-called Poincar´e rank p to its minimal integer value was investigated. How-ever, given our perturbed system [ A σ A ], it seems more plausible to reduce h , rather than p , to itsminimal integer value, since a system with h ≤ µ ε ( A ) cannot be computed from the outset, we aim in this section to generalize Moser’scriterion to detect whether the ε -rank of a given perturbed system is minimal among all equiva-lent systems. We remark that Moser’s criterion has been generalized as well to linear functionalmatrix equations in [6], and borrowed from the theory of di ff erential systems in [24] to investi-gate e ffi cient algorithmic resolution of the perturbed algebraic eigenvalue-eigenvector problem.However, despite their utility and e ffi ciency for such univariate systems, algorithms based on thiscriterion are not considered so far over bivariate fields.In [2], we generalize the work of [9, 32] for the case σ A =
0. Herein, we establish thefollowing theorem without any restriction on σ A (see [28, Section 5.5, p. 101] and AppendixC for the proof). This yields an algorithm which takes an input system [ A σ A ], and outputs atransformation T ∈ GL n ( K ) and an equivalent system [ ˜ A σ ˜ A ] : = T [ A σ A ] which has a minimal ε -rank among all systems equivalent to [ A σ A ]. Theorem 2.
Consider system (8) given by [ A σ A ] ξ hA x p ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ Ak F , and suppose that h > and m ε ( A ) > . The polynomial θ A ( λ ) : = ξ Arank ( A ( x )) det( λ I + A ( x ) ξ A + A ( x )) | ξ A = (15) vanishes identically in λ , if and only if there exists a transformation F = TG, T ∈ GL n ( K ) ,such that the equivalent system [ ˜ A σ ˜ A ] : = T [ A σ A ] has either a strictly lower ε -rank or a leadingcoe ffi cient matrix with a strictly lower algebraic rank. Moreover, σ T , σ ˜ A ≥ σ where σ = σ A + p − h if p < σ A otherwise . Example 8.
Consider the system [ A σ A ] ξ A ∂ F = A ( x , ξ A ) F with σ A = andA ( x , ξ A ) = x ξ A x ξ x ξ A (2 x + ξ A ξ A ξ A − x . ur algorithm computes the transformation F = TG, where T = ξ T ξ T ξ T , with σ T = σ A = which results in an equivalent ε -irreducible system [ A σ ˜ A ] ξ A ∂ G = ˜ A ( x , ξ ˜ A ) G where ˜ A ( x , ξ ˜ A ) = x x ξ ˜ A x x +
10 0 ξ A − x ξ ˜ A , and σ ˜ A = . One can observe that the ε -rank is diminished by two. Example 9.
Consider the first subsystem resulting from the reduction of Example 7: [ B σ B ] ξ B x ∂ W = { (cid:34) (cid:35) + (cid:34) − − (cid:35) ξ B + (cid:34) − − + x (cid:35) ξ B + O ( ξ B ) } W . It is easy to see that the leading coe ffi cient and constant matrices coincide and that θ B ( λ ) = .Hence, the ε -rank which is equal to is minimal. To proceed in the formal reduction of Example 9, one needs to introduce a ramification in ε (resp. ξ B ). The computation of the necessary ramification will be discussed in Section 4. Toestablish the results therein, the ε -Moser invariant of equivalent scalar equations is needed. Wethus introduce it in Subsection 3.3. Before proceeding however, we discuss the case h = = h > h =
1. We consider again the system given by (8):[ A σ A ] ξ A x p ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ kA F . Let h true denote the minimal integer ε -rank which can be attained upon applying a transformationin GL n ( K ) and ω ε ( A ) s denote the ε -exponential order of [ A σ A ]. In this subsection, we give amethod to decide whether h true = h true =
0. We first apply the substitution ε = ˜ ε n + to[ A σ A ]. We denote its ˜ ε -rank by ˜ h . Since ˜ h = n + >
1, we can then apply the ˜ ε -rank reductionof Theorem 2 and we denote by ˜ h true and ω ˜ ε the true ˜ ε -rank of the resulting system and its˜ ε -exponential order respectively. Under these notations, we have: Lemma 3. If ˜ h true = then h true = .Proof. Let ω ε = (cid:96) d where (cid:96) and d are coprime natural numbers. Then, on the one hand, d , (cid:96) ≥ d < n and ω ε ≤ h true . On the other hand, ω ˜ ε = ( n + ω ε and ω ˜ ε ≤ ˜ h true (the minimal ε -rankbounds the ε -exponential order, see Corollary 2). Hence, we have, ω ˜ ε = ( n + ω ε = ( n + (cid:96) d ≥ ( n + (cid:96) n ≥ (1 + n ) (cid:96). (cid:96) (cid:44) < (1 + n ) (cid:96) ≤ ˜ ω ˜ ε ≤ ˜ h true . Lemma 3 moves the problem from the case h = h >
1. It can be restated asfollows: Within the formal reduction, whenever the case h = ε = ˜ ε n + is applied so that the reduction of Theorem 2 can be applied. After this reduction, oneof the following two cases arises: • If the ε -rank of the resulting system is less than or equal to one then h true of the originalsystem is zero, and so we stop our reduction (the ε -exponential part have been computedcompletely). If one wishes to continue reduction, then the classical Arnold-Wasow ap-proach can be employed: First, put the system in Arnold’s form [3] and then use theIwano-Sibuya’s polygon to determine the proper polynomial transformation required toarrive at an equivalent system whose ε -rank is zero [23]. The reduction then continues asexplained in Subsection 2.1.1. • Otherwise, we proceed with the reduction: If the leading coe ffi cient matrix of the ramifiedsystem has two distinct eigenvalues, then we treat turning points (if any) and apply splittinglemma. Otherwise, we proceed to Section 4.We remark that in the implementation, we first try to find whether there exist constant vectors inthe left null space of G A ( λ ) (see Appendix C). If such vectors do exist, we use them to constructa transformation which might reduce the rank. If the ε -rank remains one after all the constantvectors are exhausted, we compute a candidate for the ramification in ξ A from the characteristicpolynomial. If this candidate, after applying again the ε -rank reduction results in a system withnon-nilpotent leading coe ffi cient, we keep it. Otherwise, we use Arnold-Wasow approach. ε -Moser invariant of a scalar equation Due to the equivalence between a scalar di ff erential equation [ a σ a ] and its companion system[ A σ A ], it is natural to define and question the ε -Moser invariant of the former, in the hope togain more insight into the problem. This is the goal of this subsection which is fulfilled bygeneralizing the analogous notion discussed for unperturbed scalar linear di ff erential equationsin [32, Part IV].We consider again the singularly-perturbed linear di ff erential equations given by (9)[ a σ a ] ∂ n f + a n − ( x , ξ a ) ∂ n − f + . . . + a ( x , ξ a ) ∂ f + a ( x , ξ a ) f = , where a n ( x , ξ a ) =
1. We prove the following proposition:
Proposition 3.
Given the di ff erential equation [ a σ a ] . Let τ, ν be the smallest integers such thatval ε ( a i ( x , ξ a )) ≥ ( i − n )( τ − − ν. In other words, let κ = min { (cid:37) ∈ N | val ε ( a i ) + ( n − i ) (cid:37) ≥ , ≤ i ≤ n } (16) ν = max { ( i − n )( τ − − val ε ( a i ) , ≤ i ≤ n } (17) Then the ε -Moser invariant of the system given by the associated companion matrix is µ ε ( a ) = κ + ν n . igure 4: Geometric Interpretation Remark 2 (Geometric interpretation) . Consider again the ε -polygon N ε ( a ) of [ a σ a ] constructedin Subsection 2.1.2) in a ( U , V ) -plane. First, one can construct the straight line passing throughthe point ( n , , with the smallest integer slope κ , which stays below N ε ( a ) . It is the straightline of equation V = κ ( U − n ) . Then, one finds among all the parallel lines of slope ( κ − ,the highest straight line with an integer V-intercept which stays below N ε ( a ) . The V-interceptof the latter is − ν − n ( κ − where ν is an integer. In other words, the latter has the equationV = ( U − n )( κ − − ν . Example 10.
Let σ a = − and ξ a = x σ a ε . We consider the scalar di ff erential equation [ a σ a ] ∂ f + ( x ξ − a + x ) ∂ f + x ξ − a ∂ f + ( ξ − a + ∂ f + ( − ξ − a + x ξ − a ) ∂ f − ξ − a f = . First, we plot the points ( i , val ε ( a i )) , ≤ i ≤ n = and the ε -polygon (see Figure 4). Next, weplot the straight line passing through the point ( n , = (5 , with smallest integer slope suchthat it stays below all these point. This yields V = U − = U − and τ = . Then, we plotthe straight line of slope τ − = which stays below these points: V = U − = U − − .This yields ν = . Finally, we consider γ = ( γ , . . . , γ ) = ( − , − , − , − , − . We observe thati = n − ν = and so the equivalent system is given by: [ A σ A ] x ξ A ∂ W = ξ A x ξ A ξ A x ξ A ξ A x ξ A
00 0 0 5 ξ A x ξ A x ξ A x ξ A − x ξ A − x ξ A − x ξ A − x ξ A − x − ( x − ξ A W , with σ A = σ a = − . One can verify that the system [ A σ A ] is ε -irreducible since θ A ( λ ) = x λ . Proof. (Proposition 3) For clarlity, we set ξ : = ξ a . We define γ i = max { κ ( i − n ) , ( κ − i − n ) − ν } , ≤ i ≤ n − . γ i ≤ val ε ( a i ) and equality is attained for at least one i ≥ n − ν (otherwise ν and κ can be minimized). Let i represent the smallest integer such that γ i = κ ( i − n ), then i = n − ν . Geometrically, the γ i ’s represent a broken line dominated by the ( i , val ε ( a i )). Theywill aid in the construction of an ε -irreducible system which is equivalent to (9). In fact, let w i + = ξ γ i ∂ i f , ≤ i ≤ n − . then we have ∂ w i + = ξ − κ [ ξ w i + ≤ i < i w i + i ≤ i ≤ n − + ξ κ σ a γ i x w i + ] , w n = ξ γ n − ∂ n − f = ξ − κ ∂ n − f ,∂ w n = σ a γ n − x w n + ξ − κ ∂ n f = ξ − κ [ ξ κ σ a γ n − x w n + (cid:80) n − i = α i ( x , ε ) w i + ] . where α i ( x , ξ ) = − a i ( x , ε ) ξ − γ i = (cid:80) ∞ k ∈ Z a i , k ( x ) ξ k − γ i . Let W = ( w , . . . w n ) T and A ( x , ξ ) = σ a γ ξ κ x ξ . . .σ a γ i − ξ κ x ξσ a γ i ξ κ x σ a γ i + ξ κ xx α x α . . . x α i x α i + . . . x α n − + σ a γ n − ξ κ . Then one can verify that, [ A σ a ] x ξ κ ∂ W = A ( x , ξ ) W . It remains to prove that this system is ε -irreducible. We first remark that A ( x ,
0) has rank ν = n − i due to the linear independence of its last ν rows. In fact, it’s clear that it has ν − x ’s). Moreover, α i ( x , ξ ) = − a i ( x , ξ ) ξ − γ i = − a i ( x , ξ ) ξ − val ε ( a i ) ,and so α i ( x , (cid:44) i ∈ { , . . . , i } . Thus the last ν rows are linearly independent.23etting ξ = x σ a ε we have: θ A ( λ ) = ξ ν det( λ I + A ( x ) ξ + A ( x )) | ξ = = ξ ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ + δ ( σ a γ ) x . . .λ + δ ( σ a γ i − ) x λ + δ ( σ a γ i ) x ξ λ + δ ( σ a γ i + ) x ξ x α ξ x α ξ . . . x α i ξ x α i + ξ . . . λ + δ ( σ a γ n − ) + x α n − ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ξ = = ξ ν − n + i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ + δ ( σ a γ ) x . . .λ + δ ( σ a γ i − ) xx α x α . . . x α i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where δ = κ = α i ( x , (cid:44) i ∈{ , . . . , i } , θ A ( λ ) does not vanish identically in λ and its highest possible degree is i . It followsfrom Theorem 2 that the system is ε -irreducible and µ ε ( a ) = µ ε ( A ) = κ + ν n . Corollary 1.
Given an ε -irreducible system [ A σ A ] ξ hA x p ∂ F = A ( x , ξ A ) F. Let [ a σ a ] be thescalar equation corresponding to a companion system equivalent to [ A σ A ] . Then for κ and ν defined in (16) and (17) we have: κ = h and ν = rank ( A ) .Proof. By the ε -irreducibility of [ A σ A ] and its equivalence to the companion system, we have: κ + ν n = µ ε ( a ) = µ ε ( A ) = m ε ( A ) = h + rn Consequently, κ = h and ν = rank ( A ). Corollary 2.
Under the notations of Corollary 1, we have: κ − + ν n ≤ ω ε ( A ) ≤ κ and h − + rank ( A ) n ≤ ω ε ( A ) ≤ h . Proof.
Follows from (16), (17), and (13).
4. Computing the ε -exponential order We consider again system [ A σ A ] which is given by (8):[ A σ A ] x p ξ hA ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ kA F . (18)24n this section, we assume without loss of generality that h >
0, [ A σ A ] is ε -irreducible, and A ( x ) , A , are both nilpotent. The goal of this section is to compute the ε -exponential order ω ε ( A ) and ε -polynomials of the system, which give indispensable information within formalreduction. In particular, the former determines the ramification in ε which can lead to a systemwhose leading matrix is non-nilpotent, so that the process of formal reduction can be resumed.This leads to the recursive Algorithm 1. Eventually, we can construct fundamental matrices offormal solutions in any given subdomain. We illustrate this algorithm by examples and motivateitems for further investigation in Appendix E. Letdet( λ I − A ( x , ε ) x p ξ hA ) = λ n + α n − ( x , ξ A ) λ n − + · · · + α ( x , ξ A ) . (19)such that α n = α i ( x , ξ A ) = (cid:80) ∞ j = val ε ( α i ) α i , j ( x ) ξ jA ∈ K for i ∈ { , . . . , n } . We define the ε -polygon N ε ( A ) of [ A σ A ] by taking P ε ( A ) of Section 2.1.2 to be the union of P ( i , val ε ( α i ( x , ξ A ))for i ∈ { , . . . , n } . We can then prove the following: Theorem 3.
Consider the ε -irreducible system [ A σ A ] given by (18) with h > and (19) . Ifh > n − rank ( A ( x )) then the ε -exponential order of [ A σ A ] is given by ω ε ( A ) = max ≤ i < n ( − val ε ( α i ) n − i ) , and its corresponding ε -polynomial is given by the algebraic equationE ε ( X ) = (cid:96) (cid:88) k = x σ A · val ε ( α ik ) α i k , val ε ( α ik ) X ( i k − i ) where ≤ i < i < · · · < i (cid:96) = n denote the integers i for which ω ε ( n − i ) = − val ε ( α i ) (i.e. lie onthe edge of slope ω ε of the ε -polygon N ε ( A ) of [ A σ A ] ); and α i , val ε ( α i ) ( x ) = ξ − val ε ( α i ) A α i ( x , ξ A ) | ξ A = . The proof can be established by adapting to the parametrized setting the proofs of [8, Lemma3, Lemma 4, Proposition 1, Theorem 1] (see Appendix D). Not only does this theorem computethese invariants of the system, but it also allows a further reduction of the system as follows: Sup-pose that ω ε ( A ) = (cid:96) d where (cid:96), d are relatively prime positive integers. One can then set ˜ ε = ε / d (or equivalently ˜ ξ A = ξ / dA x σ ( d − / d ) in [ A σ A ] and perform the ˜ ε -rank reduction (the minimal ε -rank is (cid:96) ). This will lead to an equivalent system whose ε -rank is (cid:96) and whose leading coe ffi cientmatrix has at least d distinct eigenvalues. The system can thus be decoupled into subsystems oflower dimensions. By repeating these procedures for each of the resulting subsystems, we candecouple the initial system into subsystem(s) of lower dimension(s) or zero ε -rank. This leads tothe recursive algorithm of Section 5.We remark that the condition h > n − rank ( A ( x )) in Theorem 3 is non-restrictive. We can alwaysarrive to a system satisfying this condition by a generalization of [8, Lemma 5], which is basedon applying the a ffi nity ( U , V ) → ( U , d V ) for some integer d to the ε -polygon. In fact, we canprove the following lemma: In fact, we want to set ˜ ε d = ε . And so we choose an integer s such that ˜ ξ Ad = ξ A x s which is equivalent to x σ d ˜ ε d = x σ + s ε . emma 4. Consider an ε -irreducible system [ A σ A ] given by (18) with h > and r = rank ( A ( x )) .Let d be an integer such that d ≥ nh − + rn and let ˜ ξ A ˜ h x ˜ p ∂ G = ˜ A ( x , ˜ ξ A ) G . be the ˜ ε -irreducible di ff erential system obtained by the ramification ε = ˜ ε d (or equivalently ˜ ξ A = ξ / dA x σ A ( d − / d and σ ˜ A = σ A ( d − ) and performing ˜ ε -rank reduction. Then µ ˜ ε ( ˜ A ) = ˜ h + ˜ rn ,where ˜ r = rank ( ˜ A ( x )) and ˜ h + ˜ r > n.Proof. By Corollary 2 we have, h − + rn ≤ ω ε ( A ) ≤ h . And due to ε = ˜ ε d , we have ω ε ( A ) = ω ˜ ε ( ˜ A ) d .Hence, ω ˜ ε ( ˜ A ) ≥ d ( h − + rn ) ≥ n . But ˜ h ≥ ω ˜ ε ( ˜ A ) and ˜ r ≥
1, which yields ˜ h + ˜ r ≥ ω ˜ ε ( ˜ A ) + ≥ d ( h − + rn ) + ≥ n + > n .
5. Formal reduction algorithm
With the algorithms of Splitting lemma, turning point resolution, ε -rank reduction, and as canbe verified by Algorithm 1 below, we have re-established constructively the well-known generalform for a fundamental matrix of formal outer solutions for an input system [ A ε ] given by (3): F = ( ∞ (cid:88) k = Φ k ( x / s ) ( x σ ε ) k / d ) exp( (cid:90) Q ( x / s , ε − / d )) , (20)where s , d are positive integers; σ is a nonpositive rational number; Q is a diagonal matrix whoseentries are polynomials in ε − / d with coe ffi cients in C (( x / s )). We refer to Q : = (cid:82) Q as the ε - exponential part (logarithms in a root of x might arise as a result of integration); and the entriesof the Φ k ( x / s )’s are root-meromorphic in x (see [36, Introduction] or [35]). Remark 3.
Under the notations and statements of Theorem 3, the leading term of Q is given by ε ω ε ( A ) (cid:90) diag( X ( x ) , . . . , X deg ( E ε ) ( x ) , , . . . , dx , where the X i ’s denote the roots of the ε -polynomial E ε ( X ) . A similar statement can be stated interms of the eigenvalues of the leading matrix coe ffi cient, x p , and ξ hA . We sum up our main results in the formal reduction algorithm, Algorithm 1, which computesthe ε -exponential part and consequently a fundamental matrix of formal solutions (20) in a givensubdomain. We recall that: • If n = h terms; • If h ≤ mini I solde , or L indalg ).
6. Conclusion and further investigations
In this article, we give an algorithm, implemented in M aple , which computes a fundamentalmatrix of outer formal solutions for singularly-perturbed linear di ff erential systems in a neigh-borhood of a turning point. The subprocedures discussed are stand-alone algorithms (Splitting, The package is available at: http : // . spec f un . inria . f r / smaddah / Research . html lgorithm 1 E xp param ( h , p , σ A , A ( x , ξ A )) : Computes the ε -exponential part of system [ A σ A ](performs formal reduction). Input: h , p , σ A , A ( x , ξ A ) of system (8) ( p = σ A = Output: Q ( x / s , ε − / d ) Q ← diag(0 , . . . while h > n (cid:44) doif A , has at least two distinct eigenvalues then Apply the ε -block diagonalization of Section 2.3;E xp param ( h , p , σ A , ˜ A ( x , ε )); Update Q and σ A ;E xp param ( h , p , σ A , ˜ A ( x , ε )); Update Q and σ A ; else if A , has one non-zero eigenvalue then Update Q from the eigenvalues of A , ; A ( x , ξ A ) ← perform eigenvalue shifting (14); ( A , is now nilpotent);E xp param ( h , p , σ A , A ( x , ξ A )); Update Q ; else if A ( x ) is not nilpotent then A ( x , ξ A ) , p , σ A , ← apply turning point resolution of Subsection 3.1; ( A , is now non-nilpotent and σ A is updated);E xp param ( h , p , σ A , A ( x , ξ A )); Update Q ; else A ( x , ξ A ) , h ← ε -rank reduction of Section 3.2; if h > h true > thenif A , has at least two distinct eigenvalues then Apply the ε -block diagonalization of Section 2.3;E xp param ( h , p , σ A , ˜ A ( x , ε )); Update Q and σ A ;E xp param ( h , p , σ A , ˜ A ( x , ε )); Update Q and σ A ; else if A , has one non-zero eigenvalue then Update Q from the eigenvalues of A , ; A ( x , ξ A ) ← perform eigenvalue shifting (14); ( A , is now nilpotent);E xp param ( h , p , σ A , A ( x , ξ A )); Update Q ; else if A ( x ) is not nilpotent then A ( x , ξ A ) , p , σ A , ← apply turning point resolution of Subsection 3.1; ( A , is nownon-nilpotent and σ A is updated);Update Q from the eigenvalues of A , ; A ( x , ξ A ) ← perform Eigenvalue shifting; ( A , is now nilpotent);E xp param ( h , p , σ A , A ( x , ξ A )); Update Q ; else Use Theorem 3 of Section 3; ω ε = (cid:96) d ; ε ← ε d ; h , A ( x , ξ A ) ← Apply ε -rank reduction of Section 3.2 ( h ← (cid:96) );E xp param ( h , p , σ A , A ( x , ξ A )); Update Q ; end ifend ifend ifend whilereturn (Q). 27ank reduction, and resolution of turning points). The formal reduction algorithm is based on thegeneralization of the algorithm given in [8] for the unperturbed counterparts of such systems.Our results are presented in the formal setting. However, the growth of the order of poles in x is tracked within the formal reduction. This gives information on the restraining index of the sys-tem, and consequently, an adequate stretching transformation can be chosen and performed. Thisfurnishes the first step in resolving Iwano’s first problem and computing, for an input system, [ ρ ]which is given by (11): [ ρ ] 0 = ρ < ρ < ρ < · · · < ρ m . In Appendix E, we try to motivate the employment of our proposed formal reduction algo-rithm to resolve the same problem for a general system: Suppose that we start the reduction withthe input system [ A σ A ], σ A =
0, given by (8) as an input. Then, upon applying Algorithm 1, wecan determine the outer solutions and the final restraining index σ f inal which allows full reduc-tion. If the restraining index is nonzero then we have a turning point and ρ m = − /σ f inal . Wecan then perform the stretching τ = x ε − ρ in the input system [ A σ A ], and apply to it Algorithm 1.This determines ρ . We show an example where the same process can be repeated with ρ todetermine ρ , and iteratively we reach a system whose final restraining index is zero. In futurework, we hope to investigate this approach and its correctness.Another field of investigation is the relation between the algebraic eigenvalues of the matrix A ( x , ε ) of such systems and the exponential part of the solution (see Example 21 and [28, Chap-ter 2]). One can then benefit from the existing work on fractional power series expansions ofsolutions of bivariate algebraic equations, to compute the restraining index (see, e.g. [17, 4] andreferences therein). On the other hand, it would be interesting to investigate a di ff erential-likereduction for a two-parameter perturbation of a JCF (see [28, Chapter 2] and references thereinfor the one-parameter case). In fact, one can observe that the main role in the reduction pro-cess is reserved to the similarity term of T [ A σ A ], i.e. T − A σ A T rather than T − ∂ T . Hence, fora non-di ff erential operator where T [ A σ A ] = T − A σ A T , the discussion is not expected to deviatesubstantially from the discussion presented here for a di ff erential one.The generalization of other notions and e ffi cient algorithms can be investigated as well (see,e.g. simple system [13, 11] and [28, Appendix A]).In Examples 18 and 20 we treat input systems with σ A <
0. The algorithms generalizedherein have been generalized to treat systems with an essential singularity [12] and di ff erencesystems as well(see, e.g. [1, 6]). This motivates investigating the adaptation of our proposedalgorithms at least to the di ff erence setting.There remain as well the questions which fall under the complexity, e.g. studying the com-plexity of this algorithm; the bounds on number of coe ffi cients needed in computations; devel-oping e ffi cient algorithms for computing a cyclic vector; and comparing it to the results of ourdirect algorithm.And finally, to give a full answer on the behavior of the solutions, the related problems ofconnection, matching, and secondary turning points, are to be studied as well (see, e.g. [20, 22]and references therein).
7. Acknowledgements
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Appendix A. Computing a companion block diagonal form
It is well-known that a scalar equation [ a σ a ] given by (9) can be expressed as a first-ordersystem by setting F = [ f , ∂ f , ∂ f , . . . , ∂ n − f ] T . However, the opposite direction of this transition is nontrivial although possible. The theoret-ical possibility stems from the work of [19] with the so called cyclic vectors. Consider againsystem (8) given by:[ A σ A ] ξ hA x p A ∂ F = A ( x , ξ A ) F , ξ A = x σ A ε and A , (cid:44) . We also recall that with the definition of ξ A , A ( x , ξ A ) = (cid:80) k = A k ( x ) ξ kA with A k ( x ) ∈ M n ( C [[ x ]])for all k >
0. In this appendix, we compute a companion block diagonal form for [ A σ A ], in a finitenumber of steps, in a neighborhood of a turning point. This algorithm is an adaptation of thatof [7], developed for the unperturbed counterpart of [ A σ A ]. It relies on a sequence of polynomial(shearing) transformations in x , ξ A and elementary row / column operations. • Shearing transformations: T = diag( x α ξ β A , x α ξ β A , . . . , x α n ξ β n A ) (A.1)where the α ’s and β ’s are respectively rational numbers and integers. We remark thatshearing transformations may alter σ A because of the α ’s. • Elementary transformations: We consider transformations of the form T = I n + P i , j ( a ) (A.2)where P i , j is a zero matrix except for entry in the i th row and j th column for i , j ∈ { , . . . , n } and a ∈ R such that a ( x = , ξ A = (cid:44)
0. Obviously, T ∈ GL n ( R ). The equivalent system[ ˜ A σ ˜ A ] = T [ A σ A ] is such that ˜ h = h , ˜ p = p , and σ ˜ A ≥ min( ph + σ A , σ A ). Moreover, it can beeasily verified that the e ff ect of this transformaiton on the system [ A σ A ] is as follows:30 i ( a ) : Multiplies the i th column and i th row by a and 1 / a respectively. It also adds( − ξ hA x p ∂ aa ) to the diagonal entry in the ( i , i ) th position. C i , j ( a ) : Adds to the j th column the i th column multiplied by a , adds to the i th row the j th row multiplied by − a , and adds ( − ξ hA x p ∂ a ) to the entry in the ( i , j ) th position. R i , j ( − a ) : Adds to the i th row the j th row multiplied by a , adds to the j th column the i th column multiplied by − a , and adds ( ξ hA x p ∂ a ) to the entry in the ( i , j ) th position.A careful choice of these transformations will allow us to establish the following result construc-tively: Theorem 4.
Consider the system [ A σ A ] given by (8) . Then there exists a transformation Twhich is a product of transformations of the forms (A.1) and (A.2) in a root of x, such that theequivalent system [ ˜ A σ ˜ A ] ξ ˜ h ˜ A x ˜ p ∂ F = ˜ AF has the block diagonal form: ˜ A ( x , ξ ˜ A ) = diag( ˜ A ( x , ξ ˜ A ) , . . . , ˜ A (cid:96)(cid:96) ( x , ξ ˜ A )) where the ˜ A , . . . , ˜ A (cid:96)(cid:96) are companion matrices. The proof of this theorem relies on two lemmas which we establish herein. Before pro-ceeding, we can always assume, without loss of generality, that A , = A ( x = , ξ A =
0) is inFrobenius canonical form, i.e., A , = diag( A , , . . . , A (cid:96)(cid:96) , ) (A.3)where the block submatrices A , , . . . , A (cid:96)(cid:96) , are constant matrices in companion form, of dimen-sions n ≥ n ≥ · · · ≥ n (cid:96) ≥ n + n + · · · + n (cid:96) = n ). Lemma 5.
Consider the system [ A ξ A ] given by (8) whose leading constant coe ffi cient A , is inFrobenius canonical form. Then, by a sequence of elementary operations of the form (A.2) , theequivalent system [ ˜ A σ ˜ A ] ξ h ˜ A x p ∂ F = ˜ AF has the following block decomposition: ˜ A ( x , ξ ˜ A ) = (cid:34) S ( x , ξ ˜ A ) U ( x , ξ ˜ A ) V ( x , ξ ˜ A ) W ( x , ξ ˜ A ) (cid:35) , (A.4) where • S ( x , ξ ˜ A ) is a n -square matrix in companion form:S ( x , ξ ˜ A ) = . . . s s . . . s n ; • U ( x , ξ ˜ A ) , V ( x , ξ ˜ A ) are of dimensions n × ( n − n ) and ( n − n ) × n respectively, U , = O n × ( n − n ) , V , = O ( n − n ) × n , and of the forms:U ( x , ξ ˜ A ) = . . . ... ... . . . u . . . u n − n and V ( x , ξ ˜ A ) = v . . . ... ... ... v n − n . . . ;31 W ( x , ξ ˜ A ) is a square matrix of dimension n − n ; • Furthermore, ˜ A , = A , .Proof. We first assume n >
1. Otherwise, A ( x , ξ A ) = [ a i j ] ≤ i , j ≤ n is already in form (A.4). Since A has the form (A.3), a i , i + ( x = , ξ = =
1. Hence, 1 / a i , i + ( x , ξ A ) ∈ R and 1 / a i , i + ( x = , ξ A = = Step 1
For each row number i from 1 to n − (1.1) We use 1 / a i , i + ( x , ξ A ) as a pivot and apply M i + (1 / a i , i + ( x , ξ A )) to set the entries inthe ( i , i +
1) positions to 1. We remark that the term ( − ξ hA x p ( a ) ∂ (1 / a )) is added to thediagonal entry in the ( i + , i + th position. (1.2) We now make use of these 1’s to set the entries in the ( i , j ) positions for i ∈ { , . . . , n − } , j ∈ { , . . . , n } , j (cid:44) i + j (cid:44) i + n ,we apply C i + , j ( − a i j ). We remark that the term ( + ξ hA x p ∂ a ) is added to the entry inthe ( i + , j ) position. Clearly, the rows 1 to i − Step 2
We have attained the anticipated form for the first n rows, i.e. we have shown howto construct an equivalent system whose first n rows have the described properties of (cid:104) S ( x , ξ A ) U ( x , ξ A ) (cid:105) . If n < n then we can proceed to work on the last n − n rows. (2.1) Again, we make use of these 1’s created in Step [(1 . i , j )positions for i ∈ { n + , . . . , n } and j ∈ { , . . . , n } to zeroes: For each i , j in theseranges, we apply R i , j − ( − a i j ) successively for each i from n + n with an innerloop of j from n to 2. Evidently, each such operation does not alter the form created the inner loopshld start from n because theterm comingfrom the deriva-tion settles inthe ( i , j − σ ˜ A changesw.r.t. σ A .by Step 1 for the first n rows, and by its preceeding operations within Step 2.Finally, since each of the operations performed is of the form (A.2), for x = ξ A =
0, suchan operation reduces to I n . Hence, the form of A , is preserved. Remark 4.
One can observe that by a repetitve application of Lemma 5 on the diagonal blocksof dimensions n ≥ n ≥ · · · ≥ n (cid:96) , we arrive at the aformentioned Arnold form. Lemma 6.
Consider the system [ A σ A ] given by (8) . Suppose that its leading constant coe ffi cientA , is in Frobenius canonical form and A ( x , ξ A ) is in form (A.4) . Then, by a sequence of constanttransformations, elementary operations of the form (A.2) , and shearings of the form (A.1) , theequivalent system [ ˜ A σ ˜ A ] ξ h ˜ A x ˜ p ∂ F = ˜ AF is such that has the following block diagonal form: ˜ A ( x , ξ ˜ A ) = (cid:34) ˜ S ( x , ξ ˜ A ) OO ˜ W ( x , ξ ˜ A ) (cid:35) , (A.5) where ˜ S ( x , ξ ˜ A ) is a r-square companion matrix where r ≥ n and ˜ W ( x , ξ ˜ A ) is a n − r squarematrix.Proof. If U ( x , ξ A ) and V ( x , ξ A ) of (A.4) are smiltaneously zero matrices or n = n then A ( x , ξ A )is already in the form (A.5). Otherwise we suppose that U ( x , ξ A ) is nonzero and limit ourdiscussion to this case. If V ( x , ξ A ) is nonzero as well, then we can proceed analogously. Let u j = (cid:80) ∞ k = u j , k ( x ) ξ kA for j ∈ { , . . . , n − n } then we define β = min ≤ j ≤ n − n ( val ε ( u j )) and α = min ≤ j ≤ n − n ( val x ( u j ,β )) .
32e remark that β ≥
0, and if β = α > U , is a zero matrix. Let T = diag( x α ξ β A I n , I n − n ) and [ B σ B ] = T [ A σ A ] . Then we have: B ( x , ξ B ) = (cid:34) S ( x , ξ A ) − x p + α − ξ β + hA ( α + βσ A ) I n x − α ξ − β A U ( x , ξ A ) x α ξ β A V ( x , ξ A ) W ( x , ξ A ) (cid:35) . B ( x ) = (cid:34) S ( x ) x − α ξ − β A U ( x ) O W ( x ) (cid:35) and B , = (cid:34) S , U , O W , (cid:35) where S , is a companion matrix of dimension n , U , is a companion matrix whose last rowhas at least one nonzero element, and W , is a block diagonal square matrix whose blocks are incompanion form. Let e i denote the i th row of the identity matrix I n and B i , denote the i th powerof B , . W then have: e B i , = i (cid:88) j = c i , j e j + e i + for i = , . . . , n − e B n , = n (cid:88) j = c n , j e j + n − n (cid:88) j = u j , β e n + j where c i , j ∈ C . Since there exists at least one j ∈ { , . . . , n − n } such that u j ,β (cid:44)
0, then thevectors e , e B , , e B , , e B n , are linearly independent. Hence, denoting by r the dimensionof the first block of the Frobenius form of B , , i.e. the degree of the minimal polynomial of B , , we have r > n . Hence, by putting the leading constant matrix in Frobenius form, applyingLemma (5), and the procedure described herein, we either arrive at the desired form in a finitenumber of steps. Proof. (Theorem 4) The proof can be attained by a recursive application of Lemmas 5 and 6.
Remark 5.
We remark that this method can give a cyclic vector and that [ ˜ A σ ˜ A ] is not uniquelydetermined by [ A σ A ] . Appendix B. Proofs for Section 2.3
Proof. (Proposition 1) Let f ( x , ξ ) = exp ( (cid:82) q ( x , ξ ) dx ) with q ( x , ξ ) ∈ (cid:83) s ∈ N ∗ C (( x ))(( ε / s )). Weremark first that ∀ i ∈ N , ∂ i f = P i ( q ) f where P i ( q ) = q i + · · · + ∂ i − x q
33s a polynomial in ( q , ∂ q , . . . , ∂ i − q ). Then, f ( x , ξ ) verifies (9) if and only if q ( x , ξ ) is a solutionof n (cid:88) i = a i ( x , ξ ) P i ( q ) . We denote by e the polar order of q in ξ ( ε -order), and write q ( x , ξ ) = ξ e ( s ( x ) + O ( ξ )) , q (cid:46) . Then, ∀ i ∈ { , . . . , n } , val ε ( P i ) = − ie and in fact P i ( q ( x , ξ )) = ξ − id ( s ( x ) i + O ( ξ )).Thus, n (cid:88) i = a i ( x , ξ ) P i ( q ) = n (cid:88) i = ξ ν i − id W i ( q ) , where val ε ( W i ) ≥
0. Let v ( e ) = min ≤ i ≤ n ( ν i − ie ) and I ( e ) = { i ∈ { , . . . , n } | val ε ( a i − ie ) = v ( e ) } .Hence, n (cid:88) i = a i ( x , ξ ) P i ( q ) = ξ v ( e ) ( (cid:88) i ∈ I ( e ) a i ,ν i ( x ) s i ( x ) + O ( ξ )) . Thus, q ( x , ξ ) = ξ e ( s ( x ) + O ( ξ )) , q (cid:46) (cid:80) ni = a i ( x , ξ ) P i ( q ) ≡ s ( x ) is a nonzero solution of the equation (cid:88) i ∈ I ( e ) a i ,ν i ( x ) X i ( x ) = . This equation has non trivial solutions if and only if | I ( e ) | >
1, i.e. if and only if e ∈ { e , . . . , e (cid:96) } .And then, X ( x ) is one of the nonzero roots of the associated determinant equation. Proof. (Theorem 1) We proceed in two steps:
Step 1 : We first block-diagonalize the leading matrix coe ffi cient: Suppose that there exists T ( x ) ∈ Gl n ( C [[ x ]]) s.t. the substitution F = T ( x ) G in [ A σ A ] yields an equivalent system[ ˜ A σ A ] ξ hA x p ∂ G = ˜ A ( x , ξ A ) G where ˜ A ( x , ξ A ) = (cid:80) ∞ k = ˜ A k ( x ) ξ kA , ˜ A k ( x ) ∈ M n ( C [[ x ]]) for all k ≥ ffi cient matrix ˜ A ( x ) is non-zero and has the following block-form in accor-dance with A , : ˜ A ( x ) = (cid:34) ˜ A ( x ) OO ˜ A ( x ) (cid:35) . Then we have:
A T ( x ) − T ( x ) ˜ A = ξ hA x p ∂ T ( x )which yields A k ( x ) T ( x ) − T ( x ) ˜ A k ( x ) = δ kh ∂ T ( x ) , where δ kh = k = h and δ kh = . (B.1)In particular (since h > A ( x ) T ( x ) − T ( x ) ˜ A ( x ) = . (B.2)We further assume that T ( x ) ∈ M n ( C [[ x ]]) is in the following form T ( x ) = (cid:34) I T ( x ) T ( x ) I (cid:35) . ≤ (cid:37) (cid:44) ς ≤
2, we have: A (cid:37)(cid:37) ( x ) − ˜ A (cid:37)(cid:37) ( x ) + A (cid:37)ς ( x ) T ς(cid:37) ( x ) = OA ς(cid:37) ( x ) + A ςς ( x ) T ς(cid:37) ( x ) − T ς(cid:37) ( x ) ˜ A (cid:37)(cid:37) ( x ) = O (B.3)Inserting the series expansions A ( x ) = (cid:80) ∞ i = A i , x i and ˜ A ( x ) = (cid:80) ∞ i = ˜ A i , x i in (B.3), and equatingthe power-like coe ffi cients, we get for i = A (cid:37)(cid:37) , = ˜ A (cid:37)(cid:37) , A ςς , T ς(cid:37) , − T ς(cid:37) , ˜ A (cid:37)(cid:37) , = O which are satisfied by setting ˜ A ςς , = A ςς and T ςς , = I , T ς(cid:37) , = O . And for i ≥ A ςς , T ς(cid:37) i , − T ς(cid:37) i , A (cid:37)(cid:37) , = − A ς(cid:37) i , − i − (cid:88) j = ( A ςς i − j , T ς(cid:37) j , − T ς(cid:37) j , ˜ A (cid:37)(cid:37) i − j , ) (B.4)˜ A (cid:37)(cid:37) i , = A (cid:37)(cid:37) i , + i − (cid:88) j = A (cid:37)ς i − j , T ς(cid:37) j , . (B.5)It’s clear that (B.4) is a set of Sylvester matrix equations that possess a unique solution due tothe assumption on the disjoint spectra of A , and A , (see, e.g., [5, Appendix A.1, p. 212-213]).Remarking that the right hand side depends solely on the A j , , T j , , ˜ A j , with j < i , equations(B.4) and (B.5) are successively soluble. Hence, such T ( x ) ∈ Gl n ( C [[ x ]]) can be constructed.Clearly, p T = ν T = σ T ≥ σ A . Moreover, it follows from (B.1) that ν ˜ A = p ˜ A =
0, and σ ˜ A ≥ σ A . Step 2 : Due to the first step, we can assume without loss of generality that the leading coe ffi cientmatrix A ( x ) of the system [ A σ A ] has a block-diagonal form in accordance with that of A , . Itsu ffi ces to seek T and ˜ A in the form T = (cid:80) ∞ k = T k ( x ) ξ and ˜ A = (cid:80) ∞ k = ˜ A k ( x ) ξ respectively where T k ( x ) , ˜ A k ( x ) ∈ GL n ( C [[ x ]]) for all k >
0. We set T ( x ) = I n , ˜ A ( x ) = A ( x ), and we rewrite A = ∞ (cid:88) k = A k ( x ) ξ A = ∞ (cid:88) k = B k ( x ) ξ, where B k ( x ) = x ( σ A − σ ) k A k ( x ) . Since σ A ≥ σ , it follows that B k ( x ) ∈ M n ( C [[ x ]]) for all k >
0. With the expansions in ξ , we getthe following set of ansatz for k ≥ k (cid:88) i = ( B k − i ( x ) T i ( x ) − T i ( x ) ˜ A k − i ( x )) = x p ( ∂ + σ ( k − h ) x ) T k − h , where T k − h = k < h . (B.6)It then follows from (B.6) that for k > B ( x ) T k ( x ) − T k ( x ) ˜ A ( x ) = k − (cid:88) i = B k − i ( x ) T i ( x ) − T i ( x ) ˜ A k − i ( x ) + x p ( ∂ + σ ( k − h ) x ) T k − h ( x ) − ˜ A k ( x ) + B k ( x ) .
35y inserting the claimed block forms of T k ( x ) and ˜ A k ( x ) for all k > ≤ (cid:37) (cid:44) ς ≤
2, we obtain: ˜ A (cid:37)(cid:37) k ( x ) = B (cid:37)(cid:37) k ( x ) B ςς ( x ) T ς(cid:37) k ( x ) − T ς(cid:37) k ( x ) ˜ A (cid:37)(cid:37) ( x ) = k − (cid:88) i = B ςς k − i ( x ) T ς(cid:37) i ( x ) − T ς(cid:37) k − i ( x ) ˜ A (cid:37)(cid:37) i ( x ) + x p ( ∂ + σ ( k − h ) x ) T ς(cid:37) k − h ( x ) + B ς(cid:37) k ( x ) . Hence, for every k >
1, the computation of T k ( x ) requires only lower order terms of T , B , and ˜ A .Hence, by setting T k ( x = = k > x ,we arrive at constant Sylvester matrix equations, which can be resolved successively due to thedisjoint spectrum of A , and A , , to compute T k ( x ) up to any desired order in x . Appendix C. Proofs for Section 3.2: ε -rank reduction In this section, we establish our constructive proof of the necessity of Theorem 2. For su ffi -ciency, one can consult [28, Section 5.5, p. 101]. We prove the following: Theorem 5.
Consider the system (8) given by: [ A σ A ] x p ξ hA ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ Ak Fwith h > and m ε ( A ) > and set rank ( A ( x )) = r . Suppose that the polynomial θ A ( λ ) : = ξ Ar det( λ I + A ( x ) ξ + A ( x )) | ξ A = vanishes identically in λ . Then there exists a transformation F = R ( x , ξ R ) G such that the equiv-alent system [ ˜ A σ ˜ A ] ξ ˜ A ˜ h x ˜ p ∂ G = ˜ A ( x , ξ ˜ A ) G = ∞ (cid:88) k = ˜ A k ( x ) ξ k ˜ A Gsatisfies either ˜ h < h or ˜ h = h and rank ( ˜ A ( x )) < r. Moreover, such a R ( x , ξ R ) can be constructedby [2, Algorithm 1] where it is always chosen to be a product of unimodular transformations inGL n ( C [[ x ]]) and polynomial transformations in ξ A . The proof of the theorem will be given after a set of intermediate results.
Lemma 7.
Given the system [ A σ A ] x p ξ hA ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ kA F , with r = rank ( A ( x )) . There exists a unimodular transformation U ( x ) in GL n ( C [[ x ]]) such thatthe leading coe ffi cient matrix ˜ A ( x ) of the equivalent system (10) given by [ ˜ A σ ˜ A ] ξ h ˜ A x p ∂ G = ˜ A ( x , ξ ˜ A ) G , σ ˜ A ≥ min( ph + σ A , σ A )36 as the form ˜ A ( x ) = (cid:34) ˜ A ( x ) O ˜ A ( x ) O (cid:35) where ˜ A ( x ) is a square matrix of dimension r and (cid:34) ˜ A ( x )˜ A ( x ) (cid:35) is a n × r matrix of full column rankr. Remark 6.
In practice, U ( x ) can be obtained by performing Gaussian elimination on the columnsof A ( x ) taking as pivots the elements of minimum valuation (order) in x.Proof. By Remark 7, ˜ A ( x ) = U − ( x ) A ( x ) U ( x ). Hence it su ffi ces to search a similarity trans-formation U ( x ). Since C [[ x ]] is a principal ideal domain (the ideals of C [[ x ]] are of the form x k C [[ x ]]), it is well known that one can construct unimodular transformations Q ( x ) , U ( x ) lyingin GL n ( C [[ x ]]) such that the matrix Q ( x ) A ( x ) U ( x ) has the Smith normal form Q ( x ) A ( x ) U ( x ) = diag( x β , . . . , x β r , , . . . , U (0)) (cid:44)
0, det( Q (0)) (cid:44) β , . . . , β r in Z with 0 ≤ β ≤ β ≤ · · · ≤ β r .It follows that we can compute a unimodular matrix U ( x ) in GL n ( C [[ x ]]) so that its last n − r columns form a C [[ x ]]-basis of ker ( A ( x )).As for finding a lower bound of σ ˜ A , one can see that since U ( x =
0) is invertible, we have: val x ( ˜ A k ) ≥ val x ( ˜ A k ) if k (cid:44) hval x ( ˜ A h ) + p otherwise . Thus, if p < σ ˜ A ≥ σ A + ph . Remark 7.
Consider system [ A σ A ] given by (8) . Let T ( x ) = ( I n + Q ( x )) ∈ GL n ( C [[ x ]]) such thatQ ( x ) = [ q i j ] ≤ i , j ≤ n , s . t . q n j ∈ C [[ x ]] , for r + ≤ j < nq i j = elsewhere . One can then verify that the resulting equivalent system (10) is [ ˜ A σ ˜ A ] ξ h ˜ A x p ∂ G = ˜ A ( x , ξ ˜ A ) G , σ ˜ A ≥ σ A where ˜ A ( x ) = ( I n + Q ( x )) − A ( x ) ( I n + Q ( x ))˜ A ( x ) = ( I n + Q ( x )) − A ( x ) ( I n + Q ( x )) − γ x p ∂ Q ( x );˜ A ( x ) = . . . (C.1) where γ = if h = and γ = otherwise. But, due to (15) , we can limit our interest to A ( x ) ,A ( x ) , ˜ A ( x ) , and ˜ A ( x ) solely. Hence, if h > then it su ffi ces to investigate T − A ( x , ξ σ A ) T . Weremark that if h = then the term T − ∂ T ( x ) should be taken into account. This is the reasonthe case h = is treated separately. This distinction is purely technical as explained in thefollowing remark and unlike for the Poincar´e rank p of an unperturbed system, it does not leadto a classification of regularity.
37y Lemma (7), we can suppose without loss of generality that A ( x ) is of the form A ( x ) = (cid:34) A ( x ) OA ( x ) O (cid:35) (C.2)with r independent columns and ( n − r ) zero columns. We partition A ( x ) in accordance with A ( x ), i.e. A ( x ) = (cid:34) A ( x ) A ( x ) A ( x ) A ( x ) (cid:35) , and consider G A ( λ ) = (cid:34) A ( x ) A ( x ) A ( x ) A ( x ) + λ I n − r (cid:35) . (C.3)This consideration of G A ( λ ) gives an ε -reduction criterion equivalent to θ A ( λ ). In fact, let D ( ξ A ) = diag( ξ A I r , I n − r ) where r = rank ( A ( x )). Then we can write ξ − A A ( x , ξ A ) = ND − where N : = N ( x , ξ A ) ∈ M n ( R ). Set D = D (0) and N = N ( x , G A ( λ )) = det( N + λ D ) = det( N + λ D ) | ξ A = = (det( A ( x , ξ A ) ξ A + λ I n ) det( D )) | ξ A = = (det( A ( x ) ξ A + A ( x ) + λ I n ) ξ rA ) | ξ A = = θ A ( λ ) . Thus, det( G A ( λ )) ≡ λ if and only if θ A ( λ ) does. We illustrate our progresswith the following simple example. Example 11. [2, Example 2] Consider [ A σ A ] ξ hA ∂ F = A ( x , ξ ) F with σ A = , h > , andA ( x , ξ A ) = ξ A − x ξ A (1 + x ) ξ A x x ξ A − x ξ A − x ξ A ξ A . Clearly, A ( x ) is nilpotent of rank andG A ( λ ) = x + x − x − x λ
20 2 0 λ . (C.4)We then have the following proposition: Proposition 4.
Given the system [ A σ A ] ξ hA x p ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ kA F , with r = rank ( A ( x )) . If m ε ( A ) > and det( G A ( λ )) ≡ then there exists a finite product oftriangular matrices T ( x ) = P ( I n + Q ( x )) where P is a permutation and det T ( x ) = ± , such thatfor the equivalent system [ ˜ A σ ˜ A ] ξ h ˜ A x p ∂ G = ˜ A ( x , ξ ˜ A ) G = ∞ (cid:88) k = ˜ A k ( x ) ξ k ˜ A G e have: G ˜ A ( λ ) = A ( x ) U ( x ) U ( x ) V ( x ) W ( x ) + λ I n − r − (cid:37) W ( x ) M ( x ) M ( x ) W ( x ) + λ I (cid:37) , (C.5) where ≤ (cid:37) ≤ n − r , W , W are square matrices of order ( n − r − (cid:37) ) and (cid:37) respectively, andrank ( (cid:34) A ( x ) U ( x ) M ( x ) M ( x ) (cid:35) ) = rank ( (cid:104) A ( x ) U ( x ) (cid:105) ) , (C.6) rank ( (cid:104) A ( x ) U ( x ) (cid:105) ) < r . (C.7) Moreover, σ ˜ A ≥ min( ph + σ A , σ A ) . We shall need the following Remark in the proof of the Proposition.
Remark 8. [2, Remark 6] Suppose that G A ( λ ) has the form (C.5) and there exists a transforma-tion T ( x ) ∈ GL n ( C (( x ))) such that G T [ A σ A ] ( λ ) has the formG T [ A σ A ] ( λ ) = A U U V W + λ I n − r − (cid:37) W O O ˜ W + λ I (cid:37) , where ≤ (cid:37) ≤ n − r and ˜ W is upper triangular with zero diagonal. Then, det( G T [ A σ A ] ( λ )) = λ (cid:37) det( (cid:34) A U V W + λ I n − r − (cid:37) (cid:35) ) . If det( G T [ A σ A ] ( λ )) ≡ then we have det( G T [ A σ A ] ( λ )) ≡ as well (rank of leading coe ffi cient matrixis unchanged). Hence, det( (cid:34) A U V W + λ I n − r − (cid:37) (cid:35) ) ≡ . (C.8) For a fixed (cid:37) ∈ { , . . . , n − r } we shall denote by G ( (cid:37) )0 the matrix (cid:34) A U V W (cid:35) of (C.5) .Proof. (Proposition 4) Since det( G A ( λ )) ≡ G A ( λ =
0) is singular.Let E (respectively E ) be the vector space spanned by the first r (resp. last n − r ) rows of G A ( λ = dim ( E + E ) = rank ( G A ( λ = < n . If dim ( E ) < r then setting (cid:37) = ffi ces to fulfill our claim. Otherwise, since dim ( E + E ) = dim ( E ) + dim ( E ) − dim ( E ∩ E ) < n , it follows that either dim ( E ) < n − r or dim ( E ∩ E ) >
0. In both cases, there exists at least a rowvector (cid:36) (1) ( x ) = ( (cid:36) (1)1 ( x ) , . . . , (cid:36) (1) n ( x )) with entries in C (( x )) in the left null space of G A ( λ = (cid:36) (1) j ( x ) (cid:44) r + ≤ j ≤ n . We can assume without loss of generality that (cid:36) (1) ( x ) has its entries in C [[ x ]]. Indeed, this assumption can be guaranteed by a constructionas in Remark 6. Let the constant matrix P (1) denote the product of permutation matrices which39xchange the rows of A , so that val x ( (cid:36) (1) n ( x )) < val x ( (cid:36) (1) j ( x )) , r + ≤ j ≤ n −
1, where val x denotes the x -adic valuation (order in x ). Let Q (1) ( x ) = [ q (1) i j ( x )] ≤ i , j ≤ n , s . t . q (1) n j ( x ) = − (cid:36) (1) j ( x ) (cid:36) (1) n ( x ) , for r + ≤ j < nq (1) i j = P (1) ( I n + Q (1) ( x )) is unimodular in GL n ( C [[ x ]]).Set F = F , A = A (0) , ˜ A = A ( (cid:37) ) and let A (1) be the matrix of the equivalent system ξ hA (1) x p ∂ F (1) = A (1) ( x , ξ A (1) ) F (1) obtained by the transformation F (0) = P (1) ( I n + Q (1) ( x )) F (1) . Thus, by Remark (7), G A (1) ( λ ) has the form (C.5) with (C.6) and (cid:37) = G A (1) ( λ =
0) is singular and the condition (C.7) does not occur, then onecan find, by the same argument as above a permutation matrix and a nozero vector (cid:36) (2) ( x ) in theleft null space of G (1) ( λ = Q (2) ( x ) = [ q (2) i j ( x )] ≤ i , j ≤ n , s . t . q (2) n − , j ( x ) = − (cid:36) (2) j ( x ) (cid:36) (2) n − ( x ) , for r + ≤ j < n − q (2) i j = G A (2) ( λ ) is then of the form (C.5) with (C.6) and (cid:37) = F ( s − = P ( s ) ( I n + Q ( s ) ( x )) F ( s ) where 1 ≤ s ≤ (cid:37) and Q ( s ) ( x ) = [ q ( s ) i j ( x )] ≤ i , j ≤ n , s . t . q ( s ) n − s + , j ( x ) = − (cid:36) ( s ) j ( x ) (cid:36) ( s ) n − s + ( x ) , for r + ≤ j < n − s + q ( s ) i j = . Then this process yields an equivalent matrix ˜ A ( x , ξ ˜ A ) : = A ( (cid:37) ) ( x , ξ A ( (cid:37) ) ) with (C.6) for which either(C.7) occurs or (cid:37) = n − r . But in the latter case one has, again by Remark 8, that det( A ( x )) = Example 12 (Continue Example 11) . A simple calculation shows that det( G A ( λ )) ≡ hence Ais ε -reducible. From (C.4) , for λ = , we have the singular matrixG A ( λ = = x + x − x − x . Let T = then the transformation F = TG yields the equivalent system ξξ h ˜ A ∂ G = ˜ A ( x , ξ ˜ A ) G where σ ˜ A = σ A and ˜ A ( x , ξ ˜ A ) = ξ ˜ A − x ξ ˜ A + x ) ξ ˜ A x x ξ ˜ A − x ξ ˜ A
00 2 ξ A − x ξ ˜ A . ˜ A ( λ ) has the form (C.5) with (cid:37) = and r = . In fact,G ˜ A ( λ ) = + x ) x λ − x λ . Lemma 8.
Given the system [ A σ A ] ξ hA x p ∂ F = A ( x , ξ A ) F = ∞ (cid:88) k = A k ( x ) ξ kA F . Set r = rank ( A ( x )) and suppose that m ε ( A ) > and G A ( λ ) has the form (C.5) with conditions (C.6) and (C.7) satisfied. Consider the shearing transformation S ( ξ A ) = diag( ξ A I r , I n − r − (cid:37) , ξ A I (cid:37) ) if (cid:37) (cid:44) and S ( ξ A ) = diag( ξ A I r , I n − r ) otherwise. Then F = S ( ξ A ) G yields the equivalent system [ ˜ A σ ˜ A ] ξ h ˜ A x p ∂ G = ˜ A ( x , ξ ˜ A ) G = ∞ (cid:88) k = ˜ A k ( x ) ξ k ˜ A G , for which rank ( ˜ A ( x )) < r and σ ˜ A ≥ σ A if p > σ A + p − h otherwise . Proof.
We partition A ( x , ξ A ) as follows ( we drop ( x , ξ A ) for clarity) A ( x , ξ A ) = A A A A A A A A A where A , A , A are of dimensions r , n − r − (cid:37), and (cid:37) respectively. It is easy to verify then that˜ A ( x , ξ A ) = S − AS − S − ∂ S = S − AS − ξ hA x p − diag( σ I r , O n − r − (cid:37) , σ I (cid:37) ) = A − ξ hA x p − σ I r ξ − A A A ξ A A A ξ A A A ξ − A A A − ξ hA x p − σ A I (cid:37) . Hence, the new leading coe ffi cient matrix is˜ A ( x ) = A U OO O OM M O and rank ( ˜ A ( x )) = rank ( A U ) < r . Moreover, due to the entries of the form ξ hA x p − σ A in ˜ A , σ A should be adjusted as claimed if p <
1. 41 lgorithm 2 ε -Rank Reduction of System [ A σ A ] for h > Input: h , p , σ A , A ( x , ξ A ) of [ A σ A ] Output: R ∈ GL n ( K ) and an equivalent system [ ˜ A σ ˜ A ] which is ε -irreducible. R ← I n ; h ← ε -rank of A ; U ( x ) ← Lemma 7 so that U − A ( x ) U has form (C.2); R ← RU ; A ← U − AU − ξ hA x p U − ∂ U ; Update σ A ; d = det( G λ ( A )); while d = h > doif h > then T ( x ) , (cid:37) ← Proposition 4 ; A ← T − AT − ξ hA x p T − ∂ T ; Update σ A ; S ( ξ A ) ← Lemma 8; A ← S − AS − ξ hA x p S − ∂ S ; Update σ A ; R ← RT S ; end if U ( x ) ← Lemma 7; Update σ A ; R ← RU ; A ← U − AU − ξ hA x p U − ∂ U ; d = det( G λ ( A )); h ← ε -rank of A ; end whilereturn (R, A,h). 42 xample 13 (Continue Example 12) . Let S ( ξ ˜ A ) = diag( ξ ˜ A , ξ ˜ A , , ξ ˜ A ) then G = S ( ξ ˜ A ) U yields ξ h ˜˜ A ∂ U = ˜˜ A ( x , ξ ) U where σ ˜˜ A = σ ˜ A and ˜˜ A ( x , ξ ˜˜ A ) = ξ ˜˜ A − x ξ ˜˜ A + x ) ε x x ξ ˜˜ A − x
00 2 ξ ˜˜ A ξ A − x . It is clear that the leading term ˜˜ A ( x ) has rank < = rank ( A ( x )) .Proof. (Theorem 5) By Lemma 7, we can assume that A ( x ) is in the form (C.2). Then, G A ( λ ) isconstructed as in (C.3). Since det( G A ( λ )) ≡
0, it su ffi ces to take the change of basis F = RG = T S G , where T and S are as in Propositions 4 and Lemma 8 respectively. Remark 9.
The ε -reducibility of [ A σ A ] implies that the rank of the leading coe ffi cient matrix canbe reduced (and consequently the ε -Moser rank) without necessarily reducing the ε -rank h ofthe system. If the ε -reduction criterion holds for a su ffi cient number of equivalent systems thena repetitive application of such a transformation results in an equivalent system whose leadingcoe ffi cient matrix has a zero rank, hence h can be reduced at least by one (e.g. Example 8). Appendix D. Proofs for Section 4
We give the proof of Theorem 3 after establishing a series of useful lemmas. The followingproofs are an adaptation to the parametrized setting of the proofs in [8, Lemma 3, Lemma 4,Proposition 1, Theorem 1] respectively. For clarity within these intermediate proofs, we willexpress systems in the equivalent notation [ A σ A ] ∂ F = A ( x , ε ) F where A ( x , ε ) ∈ M n ( K ), ratherthan [ A σ A ] ξ hA x p ∂ F = ˜ A ( x , ξ A ) F . Letdet( λ I − A ( x , ε )) = λ n + α n − ( x , ε ) λ n − + · · · + α ( x , ε ) . (D.1)such that α n = α i ( x , ε ) = (cid:80) ∞ j = val ε ( α i ) α i , j ( x ) ε j ∈ K for i ∈ { , . . . , n } . We define the ε -polygon N ε ( A ) of [ A σ A ] as in Section 2.1.2, by taking P ε ( A ) to be the union of P ( i , val ε ( α i ( x , ε ))for i ∈ { , . . . , n } . We thus prove the following theorem, of which Theorem 3 is a straightforwardcorollary. Theorem 6.
Consider the ε -irreducible system [ A σ A ] ∂ F = A ( x , ε ) F where A ( x , ε ) ∈ M n ( K ) ,h > , and (D.1) . If h > n − rank ( A ( x )) then the ε -formal exponential order is given by ω ε ( A ) = max ≤ i < n ( − val ε ( α i ) n − i ) . Additionally, the corresponding ε -polynomial is given by the algebraic equationE ε ( X ) = (cid:96) (cid:88) k = α i k , val ε ( α ik ) X ( i k − i ) where ≤ i < i < · · · < i (cid:96) = n denote the integers i for which ω ε ( n − i ) = − val ε ( α i ) (i.e. lie onthe edge of slope ω ε of the ε -polygon N ε ( A ) of [ A σ A ] ); and α i , val ε ( α i ) ( x ) = ε − val ε ( α i ) α i ( x , ε ) | ε = . emma 9. Let A ( x , ε ) , W ( x , ε ) be matrices in M n ( K ) and M n ( R ) respectively. Put h = max (0 , − val ε ( A ( x , ε ))) and let det( λ I − A ( x , ε )) − det( λ I − A ( x , ε ) + W ( x , ε )) = α n − ( x , ε ) λ n − + α n − ( x , ε ) λ n − + · · · + α ( x , ε ) , where α n − ( x , ε ) , α n − ( x , ε ) , . . . , α ( x , ε ) lie in K . Thenval ε ( α n − i ) ≥ (1 − i ) h ≤ i ≤ n . Proof.
For any i ∈ { , . . . , n } , it follows from Cramer’s rule that α n − i = γ (cid:88) l = ( w l i − (cid:89) s = a l , s )where for 0 ≤ l ≤ γ, ≤ s ≤ i − , w l are entries in W ( x , ε ) and a l , s are entries in W ( x , ε ) or A ( x , ε ). Consequently, val ε ( α n − i ) ≥ ( i − val ε ( A ( x , ε )) ≥ h . Lemma 10.
Let A ( x , ε ) , B ( x , ε ) be two matrices in M n ( K ) such that val ε ( A ( x , ε )) ≤ val ε ( B ( x , ε )) .Consider the two systems [ A σ A ] ∂ F = A ( x , ε ) F and [ B σ B ] ∂ G = B ( x , ε ) G. Suppose that thereexists T ∈ GL n ( K ) such that [ B σ B ] = T [ A σ A ] . Put h = max (0 , − val ε ( A ( x , ε ))) and let det( λ I − A ( x , ε )) = λ n + α n − λ n − + α n − λ n − + · · · + α det( λ I − B ( x , ε )) = λ n + β n − λ n − + β n − λ n − + · · · + β . Then, we have: val ε ( α n − i ( x , ε ) − β n − i ( x , ε )) ≥ (1 − i ) h , ≤ i ≤ n . Proof.
By [31, Lemma 1] we can write T ( x , ε ) = P ( x , ε ) ε γ Q ( x , ε ) where P , Q ∈ GL n ( R ) and ε γ = diag( ε γ , . . . , ε γ n ) for some integers ( γ ≤ γ ≤ · · · ≤ γ n ). Consider [ ˜ A σ ˜ A ] : = P [ A σ A ] and[ ˜ B σ ˜ B ] : = Q − [ B σ B ]. Then we have˜ B = ε − γ ˜ A ε γ , val ε ( ˜ A ) = val ε ( A ) , and val ε ( ˜ B ) = val ε ( B ) . It follows that, det( λ I − ˜ A ) = det( λ I − P − AP + P − ∂ P ) = det( λ I − A + ( ∂ P ) P − )det( λ I − ˜ B ) = det( λ I − QBQ − + Q ∂ Q − ) = det( λ I − B + ( ∂ Q − ) Q )det( λ I − ˜ B ) = det( λ I − ε − γ ˜ A ε γ ) = det( λ I − ˜ A ) . Since P , Q , P − , Q − , are units of M n ( R ), it follows that ( ∂ P ) P − and ( ∂ Q − ) Q inherit this prop-erty as well. The rest of the proof follows as a consequence of Lemma 9. Proposition 5.
Consider the system [ A σ A ] ∂ F = A ( x , ε ) F where A ∈ M n ( K ) and let: det( λ I − A ( x , ε )) = λ n + α n − ( x , ε ) λ n − + α n − ( x , ε ) λ n − + · · · + α ( x , ε ) . et [ C σ C ] ∂ G = C ( x , ε ) G whereC ( x , ε ) = Companion ( c i ( x , ε )) ≤ i ≤ n − = . . .
00 0 1 . . . ... ... ... ... ... . . . c c c . . . c n − , (D.2) be a companion system which is equivalent to [ A σ A ] over GL n ( K ) . Then we have,val ε ( α i − c i ) ≥ (1 − ( n − i )) h ≤ i ≤ n − . Proof.
Let [ c σ c ] with c n = C σ C ] ( σ c = σ C ). Consider κ of [ c σ c ] as defined in Section 3.3. We define β i = ( n − i ) κ, for i ∈ { , . . . , n − } and [ D σ D ] = ε β [ C σ C ]. Then D ( x , ε ) = ε − β C ( x , ε ) ε β where ε β = diag( ε β , . . . , ε β n − ). It follows that D ( x , ε ) = ε − κ Companion ( ε β i c i ( x , ε )) ≤ i ≤ n − . We have, val ε ( D ( x , ε )) ≥ − κ since val ε ( ε β i c i ) = ( n − i ) κ + val ( c i ) ≥
0. By the equivalence between [ A σ A ] and [ C σ C ] (resp. [ D σ D ]) we have h ≥ κ (resp. val ε ( D ( x , ε )) ≥ − κ ≥ − h ). Letdet( λ I − D ( x , ε )) = λ n + d n − ( x , ε ) λ n − + · · · + d ( x , ε ) . Hence, by Lemma 10, we have val ε ( α i − d i ) ≥ (1 − ( n − i )) h , ≤ i ≤ n . Moreover, by Lemma 9 val ε ( d i − c i ) ≥ (1 − ( n − i )) max (0 , − val ε ( D ( x , ε ))) ≥ (1 − ( n − i )) κ ≥ (1 − ( n − i )) h . It follows that val ε ( α i − c i ) ≥ min ( val ε ( α i − d i ) , val ε ( d i − c i )) ≥ (1 − ( n − i )) h . Proof. (Theorem 6) Let [ C σ C ] ∂ G = C ( x , ε ) G be as in Proposition 5. Due to their equivalence,[ A σ A ] and [ C σ C ] have the same ε -exponential order and ε -polynomial. Hence, we have ω ε : = ω ε ( A ) = ω ε ( C ) = max ≤ i < n ( − val ε ( c i ) n − i ) and E ε ( X ) = (cid:96) (cid:88) k = c i k , val ε ( c ik ) X ( i k − i ) where 0 ≤ i < i < · · · < i (cid:96) = n denote the integer i for which ω ε ( n − i ) = − val ε ( c i ). ByCorollary 2 and Proposition 5 one has val ε ( α i − c i ) ≥ ( i − n + h = ( i − n ) ω ε + ( i − n )( h − ω ε ) + h ≥ ω ε ( i − n ) + ( − n )(1 − rn ) + h ≥ ω ε ( i − n ) + r + h − n . It follows from h + r − n > val ε ( α i − c i ) > ( i − n ) ω ε , ≤ i ≤ n .45 If i ∈ { i , . . . i (cid:96) } then ω ε ( i − n ) = val ε ( c i ) and val ε ( α i − c i ) > val ε ( c i ). Hence, val ε ( α i ) = val ε ( c i ) and α i , val ε ( α i ) = c i , val ( c i ) . • Else val ε ( α i − c i ) > ω ε ( i − n ) and val ε ( c i ) > ω ε ( i − n ). Thus, val ε ( α i ) ≥ min ( val ε ( α i − c i ) , val ε ( c i )) > ω ε ( i − n ) . This completes the proof.We also illustrate Lemma 4 with the following example:
Example 14 (Lemma 4) . Consider the system [ A σ A ] ξ A ∂ F = A ( x , ξ A ) F with σ A = andA ( x , ξ A ) = ξ A ξ A x ξ A ξ A x ξ A x ξ A x ξ A x ξ A x ξ A ξ A
00 0 x ξ A ξ A ξ A ξ A x ξ A . A ( x , ξ A ) is ε -irreducible, and we have n = , h = , and r : = rank ( A ( x )) = . Thus, thecondition h > n − r of Theorem 3 is not verified. Let us first consider a random ramification in ξ A regardless of Lemma 4. For instance, let us consider ξ A = ˜ ξ A and apply ˜ ε -rank reduction. Thisyields system [ C σ C ] ˜ ξ C ∂ G = C ( x , ˜ ξ C ) G with σ C = andC ( x , ˜ ξ C ) = x ˜ ξ C ˜ ξ C ˜ ξ C / ξ C x x ˜ ξ C ˜ ξ C / ξ C (cid:16) ˜ ξ C − x (cid:17) / ξ C x / ξ C x ξ C x ˜ ξ C ξ C x ˜ ξ C ξ C ξ C x ˜ ξ C x ˜ ξ C x ξ C ξ C , for which, ˜ h = and ˜ r = . Thus, the condition ˜ h > n − ˜ r is still not necessarily verified aftera random ramification. To guarantee that we will arrive at a system verifying this condition,we make use of Lemma 4 and choose an integer d such that d ≥ / (1 + / = / . Letd = > / , perform ξ A = ˜ ξ A in [ A σ A ] , and then ˜ ε -rank reduction. This yields the system B σ B ] ˜ ξ B ∂ G = B ( x , ˜ ξ B ) G with σ B = andB ( x , ˜ ξ B ) = ˜ ξ B x / ξ B x ˜ ξ B / ξ B (cid:16) ˜ ξ B − x (cid:17) / ξ B x / ξ B x ξ B x ˜ ξ B ˜ ξ B
00 0 ˜ ξ B ˜ ξ B x ξ B ξ B x ξ B ξ B x ξ B ˜ ξ B x x ξ B ξ B . For system [ B σ B ] , we have: ˜ h = , ˜ r = , and so the condition ˜ h = > n − ˜ r = − holds, andso, Theorem 3 can now be applied. From det( λ I − B ( x , ˜ ξ B )˜ ξ B ) , we can compute ω varepsilon ( M ) = .And so, we introduce the ramification ˜ ξ B = ˜˜ ξ B and apply ˜˜ ε -rank reduction which gives a ˜˜ ε -system of order and whose leading coe ffi cient has nonzero eigenvalues. In fact, we compute [ ˜ M ˜˜ ξ M ] ˜˜ ξ M ∂ W = ˜ M ( x , ˜˜ ξ M ) W with σ M = and ˜ M ( x , ˜˜ ξ M ) = ˜˜ ξ M x / ξ M x / ξ M − / x / ξ M x / x x ˜˜ ξ M ˜˜ ξ M
00 0 0 ˜˜ ξ M x ξ M ξ M x ξ M ξ M x ξ M x x ξ M ξ M . We remark that one can observe that this procedure was not necessary for such system sincegcd (14 , , = . Thus a simplification would leave us with a system of order . Thus, in theimplementation we first try to ignore the condition h > n − r and apply Theorem 3 directly to thesystem [ A σ A ] . For this particular example we obtain ω ε ( A ) = / . Then by performing ξ A = ˜ ξ A and the ˜ ε -rank reduction, we arrive at a ˜ ε -irreducible system of order whose leading coe ffi cient as nonzero eigenvalues. In fact, we compute [ ˜ A σ ˜ A ] ˜ ξ A ∂ G = ˜ A ( x , ˜ ξ ) G where σ ˜ A = and ˜ A ( x , ˜ ξ ˜ A ) = x ˜ ξ A ˜ ξ A / ξ ˜ A x ˜ ξ ˜ A x ˜ ξ A − / x + / ξ A / x ˜ ξ A / x ξ A ˜ ξ ˜ A x ξ A x ˜ ξ A ξ A ξ ˜ A x ˜ ξ A ξ ˜ A x x ξ A ξ ˜ A . If Theorem 3 does not lead to the desired result, we then resort Lemma 4. In the case of unper-turbed system, it is also a matter of discussion whether this condition is necessary. M. Miyakerecently claimed to have such an example (where this condition is necessary).
Appendix E. Examples
We treat with our algorithm examples from literature. For the computation of the full the ε -exponential parts, we refer to our M aple package P aram I nt [30]. Example 15 (Continue Example 9) . We resume the computation of the outer solution of system [ B σ B ] given by: [ B σ B ] ξ B x ∂ W = B ( x , ξ B ) W = { (cid:34) (cid:35) + (cid:34) − − (cid:35) ξ B + (cid:34) − − + x (cid:35) ξ B + O ( ξ B ) } W , with σ B = − . The leading matrix coe ffi cient (cid:34) (cid:35) is nilpotent and ε -irreducible. We computethe characteristic polynomial of B ( x ,ξ B ) ξ B x : λ − ( ξ B x − ξ B x λ + ( ξ B − ξ B x − ξ B x . Then the ε -formal exponential order is ω ε = and the ε -polynomial is X + x = . Thus,by setting ε = ˜ ε , ξ B = ˜ ξ B x ( ˜ ξ B = x − ˜ ε ) and applying ˜ ε -rank reduction of Section 3.2 via diag(1 , ˜ ξ B ) , we get an ˜ ε -irreducible system whose leading matrix coe ffi cient is (cid:34) − x − (cid:35) . Andso, by the turning point algorithm of Section 3.1, we apply the transformation (cid:34) x − / (cid:35) which yields (note that we can also express the former using a change of independent variablex = t ) [ ˜ B σ ˜ B ] ˜ ξ B x / ∂ U = ˜ B ( x , ˜ ξ ˜ B ) U where σ ˜ B = − and B ( x , ˜ ξ ˜ B ) = (cid:34) − (cid:35) + (cid:34) − x /
00 0 (cid:35) ˜ ξ ˜ B + (cid:34) x (cid:35) ˜ ξ B + (cid:34) x / − x / + x / (cid:35) ˜ ξ B + O ( ˜ ξ B ) . The leading matrix has two distinct eigenvalues. Applying Splitting Lemma we get [ ˜˜ B σ ˜˜ B ] ˜ ξ B x / ∂ R = ˜˜ B ( x , ˜ ξ ˜˜ B ) R where σ ˜˜ B = − and ˜˜ B ( x , ˜ ξ ˜˜ B ) = (cid:34) − i i (cid:35) + (cid:34) − x / − x / (cid:35) ˜ ξ ˜˜ B + O ( ˜ ξ B ) . Thus this system can be decoupled to any desired precision. The solution follows by straightfor-ward integration. Remark that, as expected, the system has the same ε -formal exponential orderand ε -polynomials, as its equivalent scalar equation given in Example 2. The leading term ofthe ε -exponential part Q is then given by: exp( − i √ x ε / i √ x ε / ) . Moreover, since σ ˜˜ B = − , weset ρ = / . Further computations yield the full exponential part of outer solutions of the inputsystem [ A σ A ] of Example 9: x ε − x ε and − t ε / − t ε + t ε / where x = − t . Moreover, since σ ˜˜ B = − , we set ρ = / . For the initial system of Example 9 given by: ε ∂ F = ε x F , we have obtained so far outer formal solutions. To compute inner (and probably intermediate)solutions, we set τ = x ε − ρ A = x ε − / which yields: ε ∂ τ F = ε τε F , which can be expressed equivalently as: [ E σ E ] ˜ ξ E ∂ τ F = E ( τ, ˜ ξ E ) F = ξ E τ ˜ ξ E F , with σ E = and ξ E = ε = ˜ ε = ˜ ξ E . By Algorithm 1, one can compute the transformation F = TG whereT = ˜ ξ E ξ E
00 0 1 . The resulting system is [ ˜ E σ ˜ E ] ˜ ξ E ∂ τ G = ˜ E ( x , ˜ ξ ˜ E ) G = τ G , with σ ˜ E = . ne can verify that ˜ E , has three distinct roots and so the system can be decoupled into threescalar equations. The leading term of the ε -exponential part in the inner subdomain is then givenby: exp( (cid:82) τ dz + ...ε / (cid:82) τ − + i / √ + ... dz ε /
00 0 (cid:82) τ − − i / √ + ... dz ε / ) . And further computations show that the diagonal entries of the ε -exponential part of a funda-mental matrix of formal solutions in the inner subdomain are given by: τ + (1 / τ + (1 / τ + O ( τ ) ε / and τ RootOf( z + z + + (1 / τ − (1 / z + z + + τ + O ( τ ) ε / . Since σ ˜ E = , the reduction stops here. Example 16 (Iwano-Sibuya polygon) . Consider the following scalar equation whose σ a = : [ a σ a ] ∂ f − ε − (2 x + ε ) ∂ f + ε − x f = , (E.1) One can verify that ω ε = . The Iwano-Sibuya polygon is given by the following set of points:P ω ε ( a ) = (2 , − , P = (0 , ) , P = (0 , ) , and P = ( , leading to only one slope givenby σ = − . Then the behavior can be investigated with the help of [ ρ ] 0 < ρ = − /σ = / . Hence, there is only one stretching transformation to consider: τ = x ε − / . We now wishto compute formal solutions and the sequence [ ρ ] with the matricial representation using ouralgorithm: Let F = [ f , ε ∂ f , ε ∂ f , ε ∂ f ] T then (E.1) can be expressed as the following firstorder di ff erential system ( σ A = ) [ A σ A ] ξ A ∂ F = A ( x , ξ A ) F = − x ξ A + x F , where ξ A = ε. (E.2) • For the turning point treatment, we set x = t and then apply the transformationT = t t t which yields the system [ ˜ A σ ˜ A ] ξ A t ∂ t G = ˜ A ( t , ξ ˜ A ) G = − ξ A + ξ A t / − ξ A − t − ξ A
00 0 2 t G . (E.3)50 here σ ˜ A = − / , i.e. ξ ˜ A = t − / ε .Applying the Splitting lemma decouples (E.3) into two subsystems (truncated at order inboth t and ε ) [ S σ S ] ξ S t ∂ t U = S ( t , ξ S ) U and [ S σ S ] ξ S t ∂ t U = S ( t , ξ S ) U where σ S = σ S = − / , ξ S = ξ S = t − / ε , andS ( t , ξ S ) = − / ξ S (cid:16) ξ S t + t + (cid:17) − ξ S t + ξ S t / − ξ S t − t + − / ξ S (cid:16) ξ S t + t + (cid:17) whose leading coe ffi cient has two constant nonzero eigenvalues (hence the dimension ofthe system / order of the equivalent equation can be reduced). And,S ( t , ξ S ) = (cid:34) / ξ S (3 ξ S t + t − γ / t (2 ξ S t + t + γ (cid:35) , (E.4) where γ = − / t (4 ξ S t / − ξ S t − ξ S − γ = − / t ξ S (8 ξ S t / − ξ S t − ξ S − t − . Since ε t − / = ε x − / , we have ρ = / .If we wish to continue the reduction for the second subsystem, then, as usual, we firstconsider its leading coe ffi cient:S ( t ) = t t + / t . Hence, by the turning point resolution (here it su ffi ces to factorize t), the system [ S σ S ] canbe rewritten as: [ ˜ S σ ˜ S ] ξ S t ∂ t R = ˜ S ( t , ε ) Rwhere σ ˜ S = − , ξ ˜ S = t − ε , and ˜ S ( t , ξ ˜ S ) = / ξ S (cid:16) t ξ S + t − (cid:17) ˜ γ t ξ S + / t + γ where ˜ γ = − / ξ S t + t ξ S + ξ S t + γ = − / t ξ S (cid:16) − t ξ S + ξ S t − ξ S t − t − (cid:17) . The leading constant coe ffi cient has two constant eigenvalues and hence can be decoupled. • We consider again system [ A σ A ] which is given by (E.2) , and we apply the stretching x = ε / τ which yields the system [ E σ E ] . One can verify that σ E = and so with ξ E = ε = ˜ ξ E = ˜ ε , we have: [ E σ E ] ˜ ξ E ∂ τ H = − τ ˜ ξ E ξ E + τ ˜ ξ E H . (E.5)51 hen the ˜ ε -rank reduction with T = ξ E
00 ˜ ξ E ξ E yields [ ˜ E σ ˜ E ] ∂ τ V = E ( τ, ˜ ξ ˜ E ) V = ξ E + τ − τ ˜ ξ E V , σ ˜ E = , ˜ ξ ˜ E = ˜ ε. Evidently, The ε -exponential part of [ ˜ E σ ˜ E ] is zero. A formal fundamental matrix of solu-tions can be constructed following Subsection 2.1.1. Example 17 (Roo’s equation [33]) . Consider the singularly-perturbed scalar di ff erential equa-tion with σ a = ( ξ a = ε ) : [ a σ a ] ξ a ∂ f = ( x + ξ a x + ξ a ) f . (E.6) Iwano-Sibuya’s polygon of [ a σ a ] consists of three segments connecting four points [33, p. 607],P = (0 , / , P = (1 / , , P = (1 , , and P o ε = (2 , − , and thus having the three slopes, − , − , and − . Consequently, Iwano-Sibuya’s sequence [ ρ ] of positive rational numbers is: [ ρ ] 0 < ρ = / < ρ = / < ρ = . We now wish to recompute the sequence [ ρ ] with the matrix representation of [ a σ a ] using ouralgorithm. Let F = [ f , ξ a ∂ f ] T then [ a σ a ] can be rewritten as the following first order di ff erentialsystem ( σ A = ) [ A σ A ] ξ A ∂ F = A ( x , ξ A ) F = (cid:34) x + ξ A x + ξ A (cid:35) F . (E.7) • For the resolution of the turning point, let T = (cid:34) x / (cid:35) (resolution of turning point) thenF = TG yields [ ˜ A σ ˜ A ] ξ A x ∂ G = ˜ A ( x , ξ ˜ A ) = (cid:34) + ξ ˜ A + ξ ˜ A x − ξ A x / (cid:35) G (E.8) with σ ˜ A = − and ξ ˜ A = x − ε . The system [ ˜ A σ ˜ A ] can be decoupled into two scalar equationsusing the Splitting lemma and we have ρ = / . • We consider again the system [ A σ A ] and we perform the stretching τ = x ε − / and ramifi-cation ξ E = ε = ˜ ε = ˜ ξ E . This yields: [ E σ E ] ˜ ξ E ∂ τ U = E ( τ, ˜ ξ E ) U = (cid:34) τ ˜ ξ E + ˜ ξ E τ + ˜ ξ E (cid:35) U , with σ E = . et T = (cid:34) ξ E (cid:35) ( ˜ ε -rank reduction) then U = T G yields [ ˜ E σ ˜ E ] ˜ ξ E ∂ τ G = ˜ E ( τ, ˜ ξ ˜ E ) G = (cid:34) ξ ˜ E τ + τ + ˜ ξ ˜ E (cid:35) G , with σ ˜ E = . The leading coe ffi cient matrix is nilpotent. The computation of the ˜ ε -exponential ordersuggests introducing an additional ramification in ε of order . Since we already have aramification of order , to avoid a proliferation of notations, we reset ε = ˜ ε . This yields: [ ˜˜ E σ ˜˜ E ] ˜ ξ E ∂ τ G = ˜˜ E ( τ, ˜ ξ ˜˜ E ) G = ξ E τ + τ + ˜ ξ E G , with σ ˜˜ E = , ε = ξ E = ˜ ξ E = ˜ ε . Let T = (cid:34) ˜ ξ ˜˜ E
00 1 (cid:35) then G = T W yields the following ˜ ε -irreducible system: [ J σ J ] ˜ ξ J ∂ τ W = (cid:34) τ + τ + ˜ ξ J (cid:35) W , with σ J = , and ˜ ξ J = ˜ ε = ε. Let T = (cid:34) τ (cid:35) (resolution of turning point) then W = T V yields [ ˜ J σ ˜ J ] ˜ ξ J τ ∂ τ V = ˜ B ( x , ˜ ξ ˜ J ) V = (cid:34) τ + + ˜ ξ J − ˜ ξ J τ (cid:35) Vwith σ ˜ J = − , ˜ ξ ˜ J = τ − ˜ ε = τ − ε / . Clearly, system [ ˜ J σ ˜ J ] can be decoupled into two scalarequations. To find the restraining index, we rewrite τ and ˜ ε in terms of x and ε : τ − ˜ ε = τ − ε / = ( x − ε / ) ε / = x − ε / = ( x − ε ) / , and so ρ = / . • We consider again system [ A σ A ] and perform the stretching τ = x ε − / , and the ramifica-tion ε = ˜ ε . This yields: [ C σ C ] ˜ ξ C ∂ τ Z = C ( τ, ˜ ξ C ) Z = (cid:34) τ ˜ ξ C + τ ˜ ξ C + ε (cid:35) Z , with σ C = , ξ C = ε = ˜ ε = ˜ ξ C . Let T = (cid:34) ξ C (cid:35) ( ˜ ε -rank reduction) then Z = T G yields [ ˜ C σ ˜ C ] ε / ∂ τ G = (cid:34) τ ε / + τ + (cid:35) G , with σ ˜ C = and ˜ ξ ˜ C = ˜ ξ C , which can be decoupled into two scalar equations by Splitting lemma. Since σ ˜ C = , therestraining index is infinity.And so, by our techniques, we have obtained the two slopes ρ = / and ρ = / . xample 18 (([16], Example 1, Section 9.5, p. 446)) . Consider the following singularly-perturbed scalar di ff erential equation ε∂ f − ∂ f + x f = which can be rewritten as the following di ff erential first order system [ A σ A ] ξ A ∂ F = A ( x , ξ A ) F = ξ A
00 0 ξ A − x Fwhere F = [ f , ∂ f , ∂ f ] T , σ A = , and ξ A = ε . Since A ( x ) is nilpotent, Splitting lemmacannot be applied but no treatment of turning points is required. The transformation F = x
00 1 00 0 1
G yields the following ε -irreducible equivalent system: [ ˜ A σ ˜ A ] ξ ˜ A ∂ G = ˜ A ( x , ξ ˜ A ) G = − ξ ˜ A x − x ξ ˜ A − ξ ˜ A ξ ˜ A ξ ˜ A ξ ˜ A x
01 0 0 G , with σ ˜ A = and θ ˜ A ( λ ) = − λ − x. Since the leading matrix coe ffi cient is still nilpotent, we haveto compute the ε -exponential oder. The characteristic polynomial of ˜ A ( x ,ξ ˜ A ) ξ ˜ A is given by: λ + ξ ˜ A − ξ ˜ A λ + x ξ ˜ A . Consequently, the ε -exponential oder is given by: ω ε ( ˜ A ) = max { , } = . Let ε = ˜ ε and ξ = ˜ ξ then we have: [ ˜˜ A σ ˜˜ A ] ˜ ξ ∂ G = ˜˜ A ( x , ˜ ξ ) G = − ˜ ξ x − x ˜ ξ − ˜ ξ ˜ ξ ˜ ξ ˜ ξ x
01 0 0 G , with σ ˜˜ A = . The transformation G = ˜ ξ ξ
00 0 1 / /
21 0 00 − / − / U yields the following ˜ ε -irreduciblesystem: [ B σ B ] ˜ ξ B ∂ U = B ( x , ˜ ξ B ) U = ˜ ξ B x ˜ ξ B / ξ B / (cid:16) − x − (cid:17) ˜ ξ B − / ξ B x − − / ξ B x (cid:16) − x − (cid:17) ˜ ξ B − / ξ B x − / ξ B x + U , ith σ B = , ˜ ξ B = ˜ ε . The leading matrix coe ffi cient B , has three distinct roots and consequently,the system can be decoupled into three first order scalar di ff erential equations by applying theSplitting lemma. A fundamental matrix of formal solutions is then given by: Φ ( x , ε / ) exp( − x ε /
00 0 x ε / ) . where Φ ( x , ε / ) is the product of all the transformations performed including that of the Splittinglemma. Since σ B = , the origin is not a turning point for this system and we do not needstretchings. Example 19 ([ ? ]) . We consider the scalar di ff erential equation [ a σ a ] ε ∂ f − ( x + ε ) f = , with σ a = . Setting F = [ f , ∂ f ] T , we get the following di ff erential first order system [ A σ A ] ξ A ∂ F = A ( x , ξ A ) F = (cid:34) ξ A x + ξ A (cid:35) Fwith σ A = and ξ A = ε . • Since A , is nilpotent and A ( x ) is not, we first start with the treatment of the turningpoint at x = . It su ffi ces however to factorize x from A ( x ) which results in σ A = − and ξ A = x − ε . We then apply the transformation F = (cid:34) ξ A
00 1 (cid:35)
G to get the ε -irreduciblesystem: [ ˜ A σ ˜ A ] ξ A x ∂ G = ˜ A ( x , ξ ˜ A ) G = (cid:34) x ξ A ξ ˜ A x + ξ ˜ A (cid:35) Gwith σ ˜ A = − and ξ ˜ A = x − ε . Since ˜ A ( x ) is still nilpotent, we proceed to computing the ε -exponential order from the characteristic polynomial of ˜ A ( x ,ε ) ξ A x which is given by: λ − x λ − + ξ ˜ A ξ A x . Hence, ω ε ( ˜ A ) = and the ε -polynomial is given by: X − x = . Let ε = ˜ ε and ˜ ξ ˜ A = x − ˜ ε ,then with ξ ˜ A = ˜ ξ A x we have: [ ˜ A σ ˜ A ] x ˜ ξ A ∂ G = ˜ A ( x , ˜ ξ ˜ A ) G = (cid:34) x ˜ ξ A ˜ ξ A x + x ˜ ξ A (cid:35) Gwith σ ˜ A = − . We then perform G = (cid:34) ˜ ξ ˜ A
00 1 (cid:35)
U to get the ˜ ε -irreducible system: [ B σ B ] x ˜ ξ B ∂ U = B ( x , ˜ ξ B ) U = (cid:34) x ˜ ξ B ( ˜ ξ B + x + x ˜ ξ B (cid:35) U , with σ B = − and ˜ ξ B = x − ˜ ε . ow that B ( x ) has two distinct eigenvalues and B , is nilpotent, we perform U = (cid:34) x /
00 1 (cid:35)
Wto treat the turning point. Then we have: [ ˜ B σ ˜ B ] x / ˜ ξ B ∂ W = ˜ B ( x , ˜ ξ ˜ B ) W = (cid:34) x / ˜ ξ B (1 − ˜ ξ ˜ B ) 11 + x ˜ ξ B (cid:35) W , with σ ˜ B = − and ˜ ξ ˜ B = x − ˜ ε .Since ˜ B has two distinct eigenvalues, ± , one can proceed to the Splitting lemma whichdecouples the system thoroughly. From the the eigenvalues of ˜ B or the ε -polynomial, onecan read the ε -exponential part of an outer solution of this system: exp( (cid:82) x z / dz ε / − (cid:82) x z / dz ε / ) , and deduce that ρ = / . • We consider again system [ A σ A ] and perform the stretching τ = x ε − / . After introducingthe ramification ε = ˜ ε , we have: [ S σ S ] ˜ ξ S ∂ τ U = S ( τ, ˜ ξ S ) = (cid:34) ξ S τ + (cid:35) Uwith σ S = , ˜ ξ S = ˜ ε , ξ S = ˜ ξ S . The transformation U = (cid:34) ˜ ξ S
00 1 (cid:35)
V yields [ ˜ S σ ˜ S ] ˜ ξ S ∂ τ V = ˜ S ( τ, ˜ ξ ˜ S ) V = (cid:34) τ + (cid:35) V , with σ ˜ S = , ˜ ξ ˜ S = ˜ ε . Due to the nature of the eigenvalues of ˜ S , , the system [ ˜ S σ ˜ S ] can bedecoupled into two scalar equations. σ ˜ S = suggests that there are no more stretchingsto perform. • However, if we consider again system [ A σ A ] and experiment with the stretching τ = x ε − ,we get: [ S σ S ] ξ S ∂ τ U = S ( τ, ξ S ) U = (cid:34) ξ S τ ξ S + (cid:35) Uwith σ S = and ξ S = ε . By the transformation U = (cid:34) ε
00 1 (cid:35)
V and some simplification weget [ ˜ S σ ˜ S ] ∂ τ G = ˜ S ( τ, ξ ˜ S ) V = (cid:34) τ ξ S + (cid:35) V , with σ ˜ S = and ξ ˜ S = ε . Due to the nature of the eigenvalues of ˜ S , , the system [ ˜ S σ ˜ S ] canbe decoupled into two scalar equations. Its ε -exponential part is zero.So far, this matricial approach can determine the stretching τ = x ε − / but not the stretching τ = x ε − . However, both stretchings can be obtained from Iwano-Sibuya’s polygon by the treatmentof this system as a scalar equation [ ? , Last section]. xample 20 ([16], Example 1, Section 9.5, p. 453) . We consider the scalar equation ε x ∂ f + x ∂ f − ( x + ε ) f = , which can be rewritten as the following first order di ff erential system [ A σ A ] x ξ A ∂ F = A ( x , ξ A ) F = (cid:34) ξ A xx + ξ A − x (cid:35) Fwhere σ A = , ξ A = ε , and F = [ f , ∂ f ] T . We first start with the treatment of the turning point atx = since A , is nilpotent while A ( x ) is not. It su ffi ces here to factorize x which yields [ ˜ A σ ˜ A ] x ξ A ∂ F = ˜ A ( x , ξ ˜ A ) F = (cid:34) ξ A x x + ξ ˜ A − (cid:35) Fwhere σ ˜ A = − and ξ ˜ A = x − ε . Now that ˜ A , has two distinct eigenvalues, we can proceed tothe Splitting lemma. The transformation F = (cid:34) x (cid:35) G block-diagonalizes ˜ A ( x ) which yields [ ˜˜ A σ ˜˜ A ] x ξ A ∂ G = ˜˜ A ( x , ξ ˜˜ A ) G = − ξ A x − − ξ A x − ξ A x + ξ ˜˜ A ξ A x ξ A x G , where σ ˜˜ A = − and ξ ˜˜ A = x − ε and from which we can deduce that the ε -exponential part isgiven by: exp( (cid:34) − x ε + O ( ln ( x ) ε − ) 00 O ( ln ( x ) ε − ) (cid:35) ) . Since the restraining index is nonzero, we can deduce that ρ = / and apply the stretching τ = x ε − / in the input system [ A σ A ] and repeat the formal reduction procedure. Example 21.
In this example, we use our algorithm to explain some of the computations in theintroductory example in the light of the eigenvalues of the matrix of the system. Thus, we consideragain the system [ A σ A ] given by (1) : [ A σ A ] ε ∂ F = A ( x , ε ) F = (cid:34) x − ε (cid:35) F . We remark that the eigenvalues of A ( x , ε ) are given by λ , = ± ( x − ε ) / . We thus have thetwo possible expansions which correspond to the two subdomains above: • λ , = ± ( x − ε ) / = ± i ε / (1 − ε − x − ε − x + . . . ) , valid for | ε − x | < . • λ , = ± ( x − ε ) / = ± x / (1 − ε x − − ε x − + . . . ) , valid for | ε x − | < ..