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Received: 26 December 2014 / Accepted: 15 April 2015 / Published online: 8 May 2015
The local analytic theory of linear ordinary differential equations exists in two parallel flavours, either that of systems of several first order equations, or of scalar (higher order) equations. One can relatively easily transform one type of objects to the other, yet this transformation loses some additional structures.
Let $\Bbbk$ be a differential field, called the field of coefficients. We will be interested almost exclusively in the field $\M=\M(\C^1,0)$ of meromorphic germs at the origin $t=0$ on the complex line $\C=\C^1$, the quotient field of the ring $\Ox=\Ox(\C^1,0)$ of holomorphic germs at the origin. The standard $\C$-linear derivation $\d=\frac{d}{dt}$ acts on both $\Ox$ and $\M$ according to the Leibniz rule and extends on vector and matrix functions with entries in $\Bbbk$ in the natural way.
Let $A\in\Mat(n,\Bbbk)$ be an $(n\times n)$-matrix function, called the coefficients matrix, $A=\|a_{ij}(t)\|_{i,j=1}^n$, $a_{ij}\in\Bbbk$. This matrix defines the homogeneous system of linear ordinary equations
\begin{equation}\label{equation.1.1} \d x=Ax,\quad x=(x_1,\dots,x_n)\in\C^n, \quad t\in(\C,0). \end{equation} | (1) |
Alternatively, one may consider homogeneous linear ordinary differential equations of the form
\begin{equation}\label{equation.1.2} a_0\d^n u+a_1\d^{n-1}u+\cdots+a_{n-1}\d u+a_n u=0,\quad a_0,\dots,a_m\in\Bbbk,\;a_0\ne0. \end{equation} | (2) |
As before, generically solution exists only as a multivalued function defined for $t\ne 0$ and ramified over the origin.
One can easily transform the Eq. (2) to a system (1) by introducing the variables $x_k=\d^{k-1}u$, $k=1,\dots,n$. The corresponding first order identities take the form
\begin{equation}\label{equation.1.3} \d x_k=x_{k+1},\quad k=1,\dots,n-1,\quad \d x_{n}=-a_0^{-1}(a_1x_{n-1}+\cdots+a_nx_1). \end{equation} | (3) |
The group $\G=\GL(n,\Bbbk)$ of invertible matrix functions with entries in the field $\Bbbk$ acts naturally on the space of all linear systems of the form (1) . Namely, if $H=\|h_{ij}(t)\|_{i,j=1}^n$, $h_{ij}\in\Bbbk$, is such a function with the inverse $H^{-1}\in\GL(n,\Bbbk)$, then one can ``change variables'' in (1) by substituting $y=Hx$, $y=(y_1,\ldots,y_n)\in\C^n$. This substitution transforms (1) to the identity $\d y=(\d H)x+H\d x=(\d H)H^{-1}y+HAH^{-1}y$, so that
\begin{equation}\label{equation.1.5} \d y=By,\quad B\in\Mat(n,\Bbbk),\quad B= (\d H)\cdot H^{-1}+HAH^{-1}. \end{equation} | (5) |
Two systems $\d x=Ax$ and $\d y=By$ are called gauge equivalent, if there exists an element $H\in\G$ such that (5) holds. Since $\G$ is a group, this equivalence naturally is reflexive, symmetric and transitive. Thus one can formulate the problem of classification: what is the simplest form to which a given linear system can be transformed by a suitable gauge transformation? The corresponding theory is fairly well established, see below for the initial results.
Unfortunately, the notion of gauge equivalence is too restricted to deal with high order equations: indeed, since the unknown function is scalar, only the transformations of the form $u=hv$, $h\in\Bbbk$, can be considered, but one cannot expect this small group to produce a meaningful classification.
Instead it is natural to consider $\Bbbk$-linear changes of variables of a more general form which involve the unknown function and its derivatives. More specifically, one can choose a tuple of functions $h=(h_0,\dots,h_{n-1})\in\Bbbk^n$ and use it to change the dependent variable from $u$ to $v$ as follows,
\begin{equation}\label{equation.1.6} v=h_1\d^{n-1}u+h_2\d^{n-2}u+\cdots+h_{n-1}\d u+h_{n}u. \end{equation} | (6) |
The new variable $v$ also satisfies a linear differential equation which can be derived as follows (cf. with Sect. 1.2). Differentiating the formula (6) for $v$ by virtue of the Eq. (2) , one can see that all higher order derivatives $\d^i v$ can be expressed as linear combinations (over $\Bbbk$) of the formal derivatives $\d^j u$, $u=0,\dots,n-1$. The space of such combinations is $n$-dimensional, so no later than on the $n$th step there will necessary appear an identity of the form $b_0\d^m v+b_1\d^{m-1}v+\cdots+b_{m-1}\d v+b_mv=0$, $b_0\ne0$, $b_j\in\Bbbk$, $m\le n$, which is the transform of the Eq. (2) by the action of (6) . Classically, the initial equation and the transformed equation are called equations of the same type, see [Singer 1996], [Tsarev 2009] and [Ore 1933], but we would prefer to use the term ``Weyl equivalence'' (justifying the fact), with an intention to refine it by imposing additional restrictions on the transformation (6) .
In order for this change of variables to be ``faithful'', one has to impose the additional condition of nondegeneracy: no solution of (2) is mapped into identical zero by the transformation (6) . Indeed, if this extra assumption is violated, one can easily transform the initial equation to the trivial (meaningless) form $0=0$. On the other hand, accepting this condition guarantees (as can be easily shown) that the transformed equation has the same order $m=n$.
Still a few questions remain unanswered by this naïve approach. The transformation (6) , unlike the gauge transformation of linear systems, is rather problematic to invert: transition from $u$ to $v$ always has a nontrivial kernel (solutions of the corresponding homogeneous equations). In addition, ``restoring'' $u$ from $v$ is in general a transcendental operation requiring integration of linear equations, and it is by no means clear how one should proceed.
The algebraic nature of these questions was studied since 1880s by F. Frobenius, E. Landau, A. Loewy, W. Krull and culminated in the perfect form in the brilliant paper by [Ore 1933]. The idea is to consider the noncommutative algebra of differential operators $\Bbbk[\d]$ with coefficients in $\Bbbk$. The next Sect. 2.1 summarizes the necessary fundamentals of the ``algebraic theory of noncommutative polynomials'' following [Ore 1933].
From this moment we focus on the special case where $\Bbbk=\M$ is the differential field of meromorphic germs at the origin and denote for brevity $\W=\M[\d]$ the algebra of operators with meromorphic coefficients.
For each linear system (1) or a high order Eq. (2) with meromorphic coefficients one can choose representatives of germs of all coefficients $a_{ij}(t)$, resp., $a_i(t)$ in a punctured neighborhood of the origin $(\C^1,0){\ssm}\{0\}$ so small that all representatives are holomorphic in this punctured neighborhood. The classical theorems of analysis guarantee that solutions of the system (resp., equation) are holomorphic on the universal cover of this punctured neighborhood, i.e., in the more traditional terminology, are multivalued analytic functions on $(\C^1,0)$ ramified at the origin.
If the coefficients of the system (1) are holomorphic at the origin, i.e., $A\in\Mat(n,\Ox)\subsetneq\Mat(n,\M)$, then for the same reasons solutions of the system are holomorphic (hence single-valued) at the origin. This case is called nonsingular, and the corresponding matrix equation admits a unique solution $X\in\GL(n,\Ox)$ with the initial condition $X(0)=E$ (the identity matrix).
Solution $X$ of a general matrix equation $\d X=AX$ with $A\in\GL(\M,n)$ after continuation along a small closed loop around the origin gets transformed into another solution $X'=XM$ of the same equation. The monodromy matrix $M\in\GL(n,\C)$ depends on $X$.
A homogeneous Eq. (2) defined by a linear operator $L=\sum_{i=0}^n a_i\d^{n-i}$ can always be multiplied by a meromorphic multiplier so that all its coefficients become holomorphic and at least one of them is nonvanishing at the origin. The reduction (3) shows that if it is the leading coefficient $a_0$ that is nonvanishing, then all solutions of the equation $Lu=0$ are holomorphic at the origin (we call such operators nonsingular), otherwise they may be ramified at the origin.
Choose a neighborhood $U=(\C^1,0)$ and meromorphic representatives of the germs $a_j(\cdot)$ which have no other poles in $U$ expect for $t=0$. If $0\ne t_0\in U$ is any other point in the domain of the system (equation), then it is well known that germs of solutions of the system (equation) $Lu=0$ form a $\C$-linear subspace in $Z_L\subset\Ox(\C,t_0)$ of dimension $\dim_\C Z_L$ exactly equal to $n$. After the analytic continuation along a small loop around the origin, this space is mapped into itself by a linear invertible map called the monodromy transformation (monodromy, for short): for any basis $u_1,\dots,u_n$ in the space of solutions (considered as a row vector function), we have
\begin{equation}\label{equation.1.7} \Delta \begin{pmatrix}u_1&\cdots& u_n\end{pmatrix}=\begin{pmatrix}u_1&\cdots& u_n\end{pmatrix}M \end{equation} | (7) |
The gauge transformation group $\G=\GL(n,\M)$ introduced above, may be too large for certain problems of analysis, see Sect. 1.4. For several reasons it is interesting to consider a smaller group $\G_h=\GL(n,\Ox)$ of holomorphic matrix functions which are holomorphically invertible. It is the semidirect product of $\GL(n,\C)$ and the group $\G_0$ of holomorphic matrix germs $H$ which are identical at the origin, $\G_0=\{H\in\G\colon H(0)=E\}$.
Besides, one can identify two types of singularities of linear systems, characterized by strikingly different behavior of solutions, called respectively regular (in full, regular singular, to avoid confusion with nonsingular systems) and irregular singularities. Recall (Ilyashenko and Yakovenko [Ilyashenko and Yakovenko 2008], Definition 16.1) that the system (1) is called regular if the norm $|X(t)|$ of any its fundamental matrix solution grows no faster than polynomially when approaching the singular point in any sector on the $t$-plane (more precisely, on the universal cover of $(\C^1,0){\ssm} 0$):
\begin{equation}\label{equation.1.8} |X(t)|\le Ct^{-N} \quad\forall t\in(\C^1,0),\ \alpha<\Arg t<\beta,\quad C>0,\ N<+\infty, \end{equation} | (8) |
The principal results on classification of linear systems are summarized in Table 1, based on Ilyashenko and Yakovenko ([Ilyashenko and Yakovenko 2008], §16, §20).
Type of Singularity / Group | Holomorphic $\G_0$ | Meromorphic $\G$ |
Nonsingular | Trivial | Trivial |
Fuchsian nonresonant Fuchsian resonant |
Euler Polynomial integrable |
Euler |
Regular non-Fuchsian | Rational | |
Irregular nonresonant Irregular resonant |
Formally diagonalizable, divergent Ramified gauge transforms are required |
Polynomial normal form.
The system takes the
form $\d X=t^{-1}(B_0+B_1t+B_2t^2+\cdots+B_pt^p)X$, where $p$ is the maximal integer
difference between the eigenvalues of the Jordan matrix $B_0$. The matrices $B_k$ may have
nonzero entry in the $(i,j)$th position only if $\l_i-\l_j=k$, that is, are very sparse.
The system in the normal form can be explicitly solved: there exists a fundamental matrix
solution of the form $X(t)=t^Pt^Q$ with two constant matrices $P,Q\in\Mat(n,\C)$
not commuting between themselves.
Rational normal norm In this case the normal form is rational and explicit but its description is off the main track of this work.
Irregular systems For irregular systems with the matrix of coefficients represented by a Laurent series $A(t)=t^{-r}(A_0+tA_1+\cdots)$, $r\ge 2$, the definition of non-resonance requires that the eigenvalues of the leading matrix coefficient $A_0$ are pairwise different. In the nonresonant case one can find a formal matrix series $H(t)=E+H_1t+H_2t^2+\cdots$ which reduces the system to a diagonal normal form $\d X=t^{-r}D(t)X$ with a diagonal polynomial normal form, with $D(0)=A_0$, but this series almost always diverges, see Ilyashenko and Yakovenko ([Ilyashenko and Yakovenko 2008], §20). To deal with the resonant case, one has to consider gauge transformations with entries being themselves ramified, i.e., involving noninteger powers of $t$. We will not deal with irregular systems or equations in this paper.
The notions of (ir)regularity can be defined also for linear equations of higher order. Somewhat mysteriously, unlike in the case of general linear systems, it is equivalent to a condition on the order of the poles of the ratios $a_i/a_0\in\M$ of the coefficients of the equation (this condition is also called the Fuchsian condition).
We study the classification of nonsingular or Fuchsian (singular) equations with respect to the Weyl equivalence (formally introduced below).
It can be easily shown (see below) that nonsingular equations are Weyl equivalent to the trivial equation $\d^n u=0$, whose solutions are polynomials of degrees $\le n-1$. An equally simple fact is the Weyl equivalence of any Fuchsian equation to an Euler equation. Furthermore, we show that the property of a Fuchsian equation to possess only holomorphic (or meromorphic) solutions can be expressed in terms of Weyl equivalence.
In our paper we introduce a more fine Fuchsian equivalence, or $\F$-equivalence for short, using expansion of operators in noncommutative Taylor series. It turns out that the corresponding classification of Fuchsian operators is very similar to the holomorphic classification of Fuchsian systems. In particular, in the nonresonant case any Fuchsian equation is $\F$-equivalent to an Euler equation, while resonant operators are $\F$-equivalent to operators with polynomial coefficients, i.e., from $\C[t][\d]$. Finally, we show that any (resonant) Fuchsian operator is $\F$-equivalent to an operator which is Liouville integrable, that is, whose solutions can be obtained from rational functions by iterated integration and exponentiation.
In this section we recall the basic facts about the algebra of differential operators with coefficients from a differential field.
Consider the $\C$-algebra $\Bbbk[\d]$ generated by the differential field $\Bbbk$ and the symbol $\d$ with the noncommutative multiplication satisfying the Leibniz rule,
\begin{equation}\label{equation.2.9} \d\cdot a=a\cdot \d+a',\quad a,a'\in\Bbbk,\quad a'=\d a=\text{ the derivative of }a. \end{equation} | (9) |
Any operator from $\Bbbk[\d]$ can be uniquely represented under the ``standard form''
\begin{equation}\label{equation.2.10} L=a_0\d^n+a_1\d^{n-1}+\cdots+a_{n-1}\d+a_n,\quad a_0,\dots,a_n\in\Bbbk,\ a_0\ne 0 \end{equation} | (10) |
The key property of the algebra $\Bbbk[\d]$ is the possibility of division with remainder. Indeed, if $n=\ord L\ge m=\ord M$, then the difference $L-a_0b_0^{-1}\d^{n-m}M$ is an operator with zero (absent) ``leading coefficient'' before $\d^{n}$, i.e., is of order strictly less than $n$. Iterating this order depression, one can find two operators $Q,R\in\Bbbk[\d]$ such that
\begin{equation}\label{equation.2.11} L=QM+R,\quad \ord Q=\ord L-\ord M,\quad \ord R<\ord M. \end{equation} | (11) |
This construction allows to define for any two operators $L,M\in\Bbbk[\d]$ their greatest common divisor $D=\gcd(L,M)$ as the operator of maximal order which divides both $L$ and $M$ (this operator is defined modulo a multiplication by an element from $\Bbbk$). The Euclid algorithm (Ore [Ore 1933], Theorem 4) guarantees that for any $L,M$ there exist $U,V\in\Bbbk[\d]$ such that
\begin{equation}\label{equation.2.12} UL+VM=\gcd(L,M),\quad \ord U<\ord M,\ \ord V<\ord L. \end{equation} | (12) |
Denote by $\W$ the local Weyl algebra $\Bbbk[\d]$ in the case where $\Bbbk=\M$ is the differential field of meromorphic germs.
If an operator $L$ is divisible by $M$ in $\W$, then their spaces of solutions $Z_L$, resp., $Z_M$, are subject to the inclusion $Z_M\subseteq Z_L$. Conversely, if for two operators $L,M\in\W$ we have $Z_M\subseteq Z_L$, then $L$ is divisible by $M$. Indeed, otherwise the remainder of division of $L$ by $M$ would be an operator of order strictly less than $\ord M$, whose solutions form the space of superior dimension $\dim Z_M=\ord M$. In terms of solutions,
\begin{equation}\label{equation.2.13} \begin{aligned} D&=\gcd(L,M)\iff Z_D=Z_L\cap Z_M, \\ P&=\lcm(L,M) \iff Z_P=Z_L+Z_M \end{aligned} \end{equation} | (13) |
Thus two equations $Lu=0$ and $Mv=0$ are of the same type in the sense of Sect. 1.3, if their order is the same and there exists an operator ${H}\in \mathscr{W}$ which maps $Z_L$ to $Z_M$ isomorphically: for any $u$ such that $Lu=0$, the function $v=Hu$ is annulled by $M$.
We will abbreviate the words ``Weyl equivalence'' (resp., conjugacy) to $\W$-equivalence (conjugacy) for simplicity.
The symmetry is less trivial, see Ore ([Ore 1933], Theorem 13). For the reader's convenience we provide here a short direct proof due to Yu. Berest. It is convenient to formulate it as a separate lemma. $\square$
An operator $L\in\W$ of the form (10) is referred to as nonsingular, if all its coefficients are holomorphic, $a_i\in\Ox(\C,0)$, and the leading coefficient is invertible, $a_0(0)\ne0$. Nonsingular operators can be reduced by the transformation (3) to a holomorphic (nonsingular) system of first order equations. An immediate conclusion is that the corresponding equation $Lu=0$ has only holomorphic solutions, and a fundamental system of solutions $\{u_k\}_{k=1}^n$ can always be chosen so that $u_k(t)=t^k+\cdots$ where the dots stand for terms of order greater than $k$.
There exists another special subclass of linear operators $L\in\W$ with the property that the respective linear equations $Lu=0$ enjoy a certain regularity, namely, all their solutions grow moderately when approaching the singular point at the origin. Unlike the general linear systems (1) , such operators admit precise algebraic description. It can be given in several equivalent forms.
Note that together with the ``basic'' derivation $\d$ any other element $a\d\in\W$ is also a derivation of the field $\M$ ($\C$-linear self-map satisfying the Leibniz rule). It can be used as the generator of the algebra $\W$. We will be mostly interested in the Euler derivation $\eu=t\d\in\W$ with the commutation rule
\begin{equation}\label{equation.2.15} \eu=t\cdot\d, \quad \eu \cdot t^m=t^m\cdot (\eu+m),\quad\forall m\in\Z, \end{equation} | (15) |
For any polynomial $w\in\C[\eu]$ in the variable $\eu$ denote by $w\sh j$, $j\in\Z$, the shift of the argument:
\begin{equation}\label{equation.2.16} w\mapsto w\sh j,\quad w\sh j(\eu)=w(\eu+j),\quad j\in\Z. \end{equation} | (16) |
\begin{equation}\label{equation.2.17} \forall w\in\C[\eu],\quad \forall j\in\Z,\quad wt^j=t^jw\sh j. \end{equation} | (17) |
Substituting $\d=t^{-1}\eu$ and re-expanding terms, any operator $L\in\W$ can be represented under the form
\begin{equation}\label{equation.2.18} L=r_0\eu^n+r_1\eu^{n-1}+\cdots+r_{n-1}\eu+r_n,\quad r_i\in\M,\; r_0\ne 0. \end{equation} | (18) |
\begin{equation}\label{equation.2.19} r_0,r_1,\dots, r_n\in\Ox(\C^1,0),\quad r_0(0)\ne 0. \end{equation} | (19) |
An operator is called Eulerian, if all coefficients $r_0,\dots,r_n\in\C$ are constant.
We will denote by $\F\subset\W$ the set of all Fuchsian operators. It is convenient to assume that holomorphically invertible germs and meromorphic germs belong in $\F$, resp., pre-$\F$ as ``differential operators of zero order''.
A Fuchsian differential equation $Lu=0$ with $L$ as in (18) can be reduced to a Fuchsian system in the sense (1.5) by slightly modifying the computation (3) : one has to introduce the new variables as follows, $x_1=u$, and then
\begin{equation}\label{equation.2.20} \eu x_k=x_{k+1},\ k=1,\dots, n-1,\quad \eu x_{n}=-r_0^{-1}(r_1 x_{n-1}+\cdots+r_n x_1) \end{equation} | (20) |
The initial results on $\W$-equivalence are completely parallel to $\G_h$-classification of nonsingular systems and $\G$-classification of regular systems: even the ideas of the proofs remain the same.
Any nonsingular equation $Lu=0$ of order $n$ always admits $n$ linear independent solutions of the form $u_k(t)=t^{k-1}(1+\cdots)$, $k=1,\dots, n$. Indeed, one should look for solutions of the companion system (3) with a suitable initial condition $x_k(0)=1$, $x_j(0)=0$ for all $j\ne k$.
A linear operator $H$ transforming solutions $v_k=t^{k-1}$ of the equation $\d^n=0$ to solutions of the equation $Lu=0$ by the formulas (6) can be obtained by the method of indeterminate coefficients: $H=h_1\d^{n-1}+\cdots+h_{n-1}\d+h_n$. The equations $Hv_k=u_k$, $k=1,\dots, n$ correspond to a system of linear algebraic equations over $\Ox$ for the unknown coefficients $h_i$: $$ \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} h_{n}&h_{n-1}&\cdots&h_1 \end{array}\right) \left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1&t&t^2&\cdots&t^{n-1} \\ & 1& 2t&\cdots&(n-1)t^{n-2} \\ & & 2 & &\vdots \\ &&&\ddots&\vdots \\ &&&&(n-1)! \end{array}\right) =\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} u_1&u_2&\cdots&u_n \end{array}\right) $$ The matrix $J$ of coefficients, the companion matrix of the tuple of solutions $v_1=1$, $v_2=t,\dots,v_n=t^{n-1}$, is holomorphic and invertible (it is upper triangular with nonzero diagonal entries). A simple inspection shows that the leading coefficient $h_1$ cannot vanish at $t=0$, hence the operator $H$ will be nonsingular. $\square$
A minor modification of this argument proves the following general result.
In other words, the (general) Weyl classification of (pre)-Fuchsian operators coincides with the classification of their monodromy matrices, very much like the meromorphic gauge classification of linear systems (1) .
It appears that a comprehensive analog of the holomorphic gauge equivalence between Fuchsian linear systems is the Fuchsian equivalence of Fuchsian operators: modulo technical details, this equivalence means the Weyl conjugacy (14) by a Fuchsian operator $H$ subject to certain nondegeneracy constraints. We start with developing the formal theory of such equivalence via noncommutative formal power series.
Together with the representation of differential operators from the ring $\W=\M[\eu]$ as polynomials in $\eu\in\W$ with coefficients in $\M$, we can expand them in convergent noncommutative Laurent series in the variable $t\in (\C^1,0)$ with (right) coefficients from the (commutative) ring $\C[\eu]$. Any operator $L\in\W$ of order $n=\ord L$ can be expanded under the form $$ L=\sum_{k=-N}^{+\infty}t^kp_k(\eu),\quad \max_k\deg_{\eu} p_k=n, \quad N<+\infty. $$ The operator is Fuchsian if and only if all powers are nonnegative and the leading coefficient $p_0$ is of the maximal degree: $L\in\F$ if and only if
\begin{equation}\label{equation.3.21} L=\sum_{k=0}^\infty t^k\,p_k(\eu),\quad p_k\in\C[\eu],\ \deg p_k\le n,\quad \deg p_0=n. \end{equation} | (21) |
Very informally, an operator with holomorphic coefficients can be considered as a small perturbation of its Eulerization. The Fuchsian condition means that this perturbation is nonsingular, i.e., it does not increase the order of the Euler part, in the same way as the nonsingularity condition means that the operator can be considered as a small nonsingular perturbation of the operator $p(\d)$.
The key tool used in this paper will be a systematic use of the Taylor expansion (21) in exactly the same way the theory of formal series with matrix coefficients of the form $H(t)=\sum_{k=0}^\infty t^kH_k$, $H\in\Mat(n,\C)$, is used in the theory of formal normal forms of vector fields (Ilyashenko and Yakovenko [Ilyashenko and Yakovenko 2008], §4 and §16). Note the difference in the algebraic nature of the noncommutativity: in the matrix case the coefficients $H_k$ commute with the variable $t$ but in general do not commute between themselves. In the operator case the polynomial coefficients $p_k\in\C[\eu]$ commute between themselves but do not commute with $t$.
Together with the convergent noncommutative Taylor series, it is convenient to introduce the class of formal Fuchsian operators.
However, the set $\F$ is not a subalgebra of $\W$: the sum of two Fuchsian operators may well be non-Fuchsian. Hence the remainder $R$ as in (11) after the incomplete division may well turn non-Fuchsian (the leading coefficient may vanish). Yet for any two given Fuchsian operators $L,M$ of degrees $n>m$ one can construct a relaxed division with remainder $L=Q'M+R'$ with $\ord Q'=n-m$ and $\ord R'=m$ and $Q',R'$ Fuchsian. Indeed, it suffices to modify the standard division with remainder $L=QM+R$ with $\ord R\le m-1$ (assuming $Q,R$ with holomorphic coefficients) and replace $Q'=Q-1$, $R'=M+R$: the latter operators will be automatically Fuchsian.
We expect that the Fuchsian classification (and its formal counterpart) for arbitrary operators from $\W$ will be a very challenging problem with the Stokes phenomenon (Ilyashenko and Yakovenko [Ilyashenko and Yakovenko 2008], §20) manifesting itself in a new way. However, everywhere below we will deal only with the $\F$-equivalence between Fuchsian operators.
Note that we dropped the condition on the order of $H,K$ which can now be higher than $n$. Besides, in this definition we replaced the condition $\gcd(H,L)=1\in\W$ from (14) by the stronger condition on the mutual primality of the respective Eulerizations.
Reflexivity is obvious: each operator $L$ is $\F$-equivalent to itself by admissible conjugacy $H=1$ (which is a zero order Fuchsian operator).
The transitivity is even simpler compared to the proof of Theorem 1: we do not replace the composition $H_2H_1$ of $\F$-conjugacies, which is always Fuchsian, by its remainder $\mod L_1$, which may be non-Fuchsian.
However, the proof of the symmetry, given in Lemma 1 relies on the possibility of representing the identical operator $1$ by a combination $1=UL+VH$ with Fuchsian operators $U,V\in\^\F$. Simple example shows that even under the stronger assumption $\gcd(\E(L),E(H))=1$, this representation is not always possible with operators of the minimal order $n-1$.
To correct the situation, one has to allow operators of above-the-minimal order.
It will be convenient to introduce the following notation:
\begin{equation}\label{equation.3.22} \forall L,H\in\F\quad \gcd\nolimits_0(L,H)=\gcd(\E(L),\E(H))\in\C[\eu]. \end{equation} | (22) |
As follows from the proof of Lemma 1, the key step is to show that if $H$ is a Fuchsian operator such that $\gcd_0(L,H)=1$, then there exist two Fuchsian operators $U,V\in\F$ such that $UL+VH=1\in\F$ and $\gcd_0(V,L)=1$. Recall that if $p,q\in\C[\eu]$ are two relatively prime polynomials of respective degrees $n,m$, then the linear Sylvester map from $\C^m\times\C^n$ to $\C^{m+n}$
\begin{equation}\label{equation.3.23} \boldsymbol S=\boldsymbol S_{p,q}\:(u,v)\mapsto pu+qv,\quad \deg u\le m-1,\;\deg v\le n-1, \end{equation} | (23) |
The following result is the analog of the implicit function theorem for differential operators.
Note that we do not assume $R$ Fuchsian, nor claim the Fuchsianity of $U$ and $V$.
Substitute the expansions for $L=\sum_0^\infty t^k p_k$ and
$M=\sum_0^\infty t^kq_k$ and the unknown operators
$U=\sum_{0}^\infty t^k u_k$, $V=\sum_0^\infty t^k v_k$,
$p_k,q_k,u_k,v_k\in\C[\eu]$ into the equation $UL+VM=R$:
\begin{align*}
&(u_0+tu_1+t^2u_2+\cdots)(p_0+tp_1+t^2p_2+\cdots) \\
&\quad+(v_0+tv_1+\cdots)(q_0+tq_1+\cdots)=r_0+tr_1+t^2r_2\ldots
\end{align*}
Using the commutation rules (16) , we reduce this operator
identity to an infinite series of identities in $\C[\eu]$,
\begin{align*}
u_0p_0+v_0q_0&=r_0, \\
u_0\sh 1 p_1+u_1p_0+v_0\sh 1 q_1+v_1q_0&=r_1, \\
u_0\sh 2p_2+u_1\sh 1p_1+u_2p_0+v_0\sh 2q_2+v_1\sh 1q_1+v_2q_0&=r_2, \\
........................ \\
\cdots+u_k p_0+v_kq_0&=r_k,\quad \forall k\ge 0.
\end{align*}
This system has a ``triangular'' form: each left hand side is the
sum of the term $u_kp_0+v_kq_0={\boldsymbol S}(u_k,v_k)$ and terms
involved shifted polynomials $u_i\sh j$, $v_i\sh j$ with $i,j
However, a simpler argument works. Expanding $U,V$ as polynomials of $\eu$ with analytic coefficients from $\Ox(\C,0)$, $$ U=\sum_k a_k(t)\eu^k,\quad V=\sum_j b_j(t)\eu^j, $$ we see that the operator equation $UL+VM=R$ reduces to a system of linear nonhomogeneous algebraic equations with respect to the unknown coefficients $a(t),b(t)$: in a symbolic way, this system can be written as $C(t)z=f(t)$, where $C(t)$ is an $(n+m)\times(n+m)$-matrix with holomorphic entries (produced from the coefficients of the operators $L$ and $M$ and their $\eu$-derivatives), and $f(t)$ is an $(n+m)$-dimensional holomorphic vector function.
One can easily see that the condition $\gcd_0(L,M)=1$ implies that the matrix $C(0)$ is nondegenerate and the system has a holomorphic solution. The formal computation amounts to the formal inversion of the corresponding matrix $C(t)$ without even explicitly writing it down.
Unfortunately, the goal of solving the equation $UL+VH=1$ in the class of Fuchsian operators cannot be achieved using only this Lemma: indeed, there is no way to ensure that the polynomial $v_0=\E(V)$ has the maximal degree equal to $\ord V$. The way out is to look for a solution of higher order.
We look for a Fuchsian solution of the equation $UL+VH=1$ in the class of operators $\ord U\le \ord H=m$, $\ord V\le \ord L=n$ as follows, $$ U=H+U_{m-1},\ V=-L+V_{n-1},\quad \ord U_{m-1}\le m-1,\;\ord V_{n-1}\le n-1. $$ Substituting these formulas into the original equation, we transform it to the equation
\begin{equation}\label{equation.3.24} U_{m-1}L+V_{n-1}H=1-[H,L], \quad [H,L]=HL-LH. \end{equation} | (24) |
Thus the equation is solvable by virtue of Lemma 2, and $$ \E(U_{m-1})\E(L)+\E(V_{n-1})\E(H)=1\in\C[\eu]. $$ In other words, $\gcd_0(V_{n-1},L)=1$. The operator $V=-L+V_{n-1}$ is Fuchsian (since $L$ is Fuchsian of order $n$), and $\gcd_0(V,L)=\gcd_0(V_{n-1},L)=1$.
This completes the proof of the symmetry of the $\F$-equivalence.
This and the next section contain the main results of the paper. They are established on the formal level, yet at the end we will show that any $\^\F$-conjugacy between convergent Fuchsian operators in fact converges.
We start by establishing an analog of the linearization theorem for nonresonant systems, cf. with the second line in Table 1.
\begin{align}\label{equation.4.25} p_0h_0&=k_0p_0, \nonumber\\ p_0\sh 1 h_1&=k_1p_0+k_0\sh 1p_1, \nonumber \\ p_0\sh 2h_2&=k_2p_0+k_1\sh 1p_1+k_0\sh 2p_2, \nonumber \\ & ........................ \nonumber \\ p_0\sh jh_j&=k_jp_0+k_{j-1}\sh 1p_1+\cdots+k_0\sh jp_j, \nonumber \\ & ........................ \end{align} | (25) |
\begin{equation}\label{equation.4.26} p_0\sh jh_j-p_0 k_j=u_j, \end{equation} | (26) |
If $L$ is nonresonant, no two roots of $p_0$ differ by a positive integer $j$, hence $\gcd(p_0,p_0\sh j)=1$ for all $j=1,2,\dots$ and any such equation is (uniquely) solvable by a suitable pair $(h_j,k_j)$ of polynomials of degree $\le n-1$. Thus the entire infinite system admits a formal solution $(H,K)$.
It remains to show that if the series for $L=\sum t^jp_j$ was convergent, so will be the series for $L$ and $K$. This can be done by the direct estimates, yet we give a general proof avoiding all computations later, in Sect. 6. $\square$
Some properties of solutions can be easily described in terms of $\F$-equivalence. Recall that a singular point of a differential equation is called apparent, if all solutions of this equation are holomorphic at this point.
One can easily show that any $n$-dimensional $\C$-linear subspace $\ell$ in $\M(\C,0)$ (resp., in $\Ox(\C,0)$) admits a basis of the germs of the form $f_i=t^{\l_i}u_i(t)$ with pairwise different integer (resp., nonnegative integer) powers $\l_i$, $u_i\in\Ox(\C,0)$ and $u_i(0)=1$. Indeed, we can start with any $\C$-basis $f_1,\dots,f_n$ in $\ell$ and normalize them so that each function has a monic leading term $t^{\l_i}(1+\cdots)$. If there are two equal powers among the initial collection, $\l_i=\l_k$, then their difference (which cannot be identically zero by linear independence) has the leading term proportional to $t^{\mu}$, $\l_i=\l_k<\mu\in\Z$. Repeating this procedure finitely many steps, one can always achive the situation when $\l_i\ne\l_k$.
Now we construct explicitly the Fuchsian operator $H=\sum t^jh_j(\eu)$ which would transform the monomials $t^{\l_i}$, $i=1,\dots,n$, to the functions $c_if_i$ for suitable coefficients $c_i\in\C$. Note that each monomial $t^{\l_i}$ is an eigenfunction for any Euler operator, in particular, $h_j(\eu)t^{\l_i}=h_j(\l_i)t^{\l_i}$, and therefore $$ Ht^{\l_i}=\f_i(t)t^{\l_i},\quad \f_i(t)=\sum_{j\ge 0} t^jh_j(\l_i). $$ The equations $Ht^{\l_i}=t^{\l_i}(c_i+c_{i1}t+c_{i2}t^2+\cdots)$ are thus transformed to the infinite number of interpolation problems, $$ h_0(\l_i)=c_i, \quad h_j(\l_i)=c_{ij},\quad i=1,\dots,n,\quad j=1,2,\dots $$ Such problems are always solvable by polynomials $h_j\in\C[\eu]$ of degree $\le n-1$, and since $c_i=h_0(\l_i)\ne 0$, we have $\gcd(h_0,p_0)=1$. By a suitable (generic) choice of the constants $c_i\ne0$, one may guarantee that $\deg h_0=n-1$, that is, $H$ is indeed a Fuchsian operator, as required for the $\F$-equivalence. $\square$Note that in both cases the normal form is maximally resonant: all differences between the roots of the Euler part are integer.
If some of the roots of the Euler part $p_0$ differ by a natural number, then the corresponding Eq. (26) may become unsolvable and in general transforming a resonant Fuchsian operator $L\in\F$ to its Euler part $\E(L)\in\C[\eu]$ is impossible, see Example 2 below. However, one can use $\F$-equivalence to simplify Fuchsian operators.
If a Fuchsian operator $H=\sum t^jh_j(\eu)$ conjugates $L$ with another operator $M=\sum t^j q_j(\eu)\in\F$, then the left hand side of the identity $p_0(\eu)H=KL$ should be replaced by \begin{align} MH&=(p_0+tq_1+t^2q_2+\cdots)(h_0+th_1+t^2h_2+\cdots) \nonumber \\ &=p_0h_0+t(q_1h_0+p_0\sh 1h_1)+\cdots\nonumber \\ &\quad+ t^j(q_j h_0+q_{j-1}h_1\sh 1+\cdots+p_0h_j\sh j)+\cdots, \end{align} and, accordingly, the Eq. (26) should be replaced by the equations
\begin{equation}\label{equation.4.28} p_0\sh jh_j-p_0 k_j+q_j h_0= v_j,\quad j=1,2,\dots, \end{equation} | (28) |
First, we use the fact that although some of the Eq. (28) may be non-solvable, they are always solvable for sufficiently large orders.
The operators $H_0,K_0$ are (usually) non-Fuchsian, since $0=\ord h_0<\ord H_0=n-1$. However, the operators $H=H_0+L$ and $K=K_0+M$ are Fuchsian, satisfy the identity $MH=M(H_0+L)=K_0L+ML=(K_0+M)L=KL$ and the nondegeneracy condition $\gcd_0(L,H)=\gcd(p_0,p_0+1)=1$ is satisfied. $\square$
The polynomial normal form established in Proposition 3, lacks any integrability properties. Yet using the same method, one can construct a Liouville integrable $\F$-normal form for any Fuchsian operator.
\begin{equation}\label{equation.4.29} M=(\eu-\l_1+r_{1})\cdots(\eu-\l_n+r_n),\quad r_{i}=r_i(t)\in\C[t],\; r_{i}(0)=0. \end{equation} | (29) |
\begin{equation}\label{equation.4.30} L=(\eu{-}\l_1+R_1)\cdots(\eu{-}\l_n+R_n), \quad R_i=R_i(t)\in\Ox(\C^1,0),\;R_i(0)=0, \end{equation} | (30) |
The normal form established by Proposition 4 has an advantage of being Liouville integrable. Each linear equation of the first order is explicitly solvable ``in quadratures''. In particular, the homogeneous equation $$ Lu=0,\quad L=\eu-\l+r(t),\qquad r\in\C[t],\ r(0)=0, $$ has a 1-dimensional space of solutions $u(t)=Ct^\l\exp \rho(t)$, where $\rho(t)=-\int \frac{r(t)}t \,\mathrm dt$ is a polynomial in $t$.
To solve the nonhomogeneous equations, the method of variation of constants can be used to produce a particular solution using operations of integration (computation of the primitive), exponentiation and the field operations in the field $\C(t)$ of rational functions (the details are left to the reader). Iterating this computation, one can find a general solution of the equation $Mu=0$ with a completely reduced operator $M$ as in (29) : if $M=L_1L_2\ldots L_n$, $\ord L_i=1$, then solution of the equation $Mu=0$ amounts to solving a chain of equations of order $1$,
\begin{equation}\label{equation.4.31} L_1 u_1=0,\,L_2 u_2=u_1,\ldots, L_nu_n=u_{n-1},\quad u=u_n. \end{equation} | (31) |
The explicit integrability of the factorized equations allows to show that the resonant Fuchsian equations, ``as a rule'', are even not $\W$-equivalent to their Euler part.
The polynomial normal forms established in the preceding section are of rather limited interest: indeed, no attempt was made to modify the lower order terms of the Taylor expansion of the resonant Fuchsian operators.
The system of Eq. (28) can be solved recursively with respect to $h_j,k_j$ even in the resonant case $\gcd(p_0,p_0\sh j)\ne 1$, provided that $q_j$ are chosen in a suitable way: the difference $v_j-q_jh_0$ should belong to the image of the Sylvester map $\boldsymbol S_j=\boldsymbol S_{p_0,p_0\sh j}$, cf. with (23) . This image consists of all polynomials of degree $\le 2n-1$ divisible by $w_j=\gcd(p_0,p_0\sh j)\in\C[\eu]$. In this section we describe possible choices for the terms $q_j$.
Denote by $\mathfrak P=\C[\eu]/\langle p_0\rangle\simeq\C_{n-1}[\eu]$ the quotient algebra: as a $\C$-space it is $n$-dimensional and can be identified with the residues modulo $p_0$, polynomials of degree $\le n-1$.
This quotient algebra in the simplest case where all $n$ roots $\l_1,\dots,\l_n\in\C$ of $p_0$ are simple, can be identified with the $\C$-algebra of functions on $n$ points $\L=\{\l_1,\dots,\l_n\}\subseteq\C\colon\mathfrak P=\{\f\colon\L\to\C\}\simeq\C\times\cdots\times\C$: any such function can be represented as the restriction of a polynomial $h\in\C_{n-1}[\eu]$ of degree $\le n-1\colon h|_\L=\f$. The functions $\f_i$ equal to $1$ at one point $\l_i\in\L$ and vanishing at all other points $\l_k\ne \l_i$, form a natural basis of $\mathfrak P$.
For any polynomial $s\in\C[\eu]$ the multiplication by $s$ is an endomorphism of the algebra $\mathfrak P$. It is invertible (automorphism of $\mathfrak P$) if and only if $\gcd(p_0,s)=1$.
The Eq. (28) induce the equations in the algebra $\mathfrak P$:
\begin{equation}\label{equation.5.32} p_0\sh j h_j+q_jh_0=v_j,\quad j=1,2,\dots \end{equation} | (32) |
\begin{equation}\label{equation.5.33} \boldsymbol P_j h_j+\boldsymbol H q_j=v_j \end{equation} | (33) |
\begin{equation}\label{equation.5.35} \boldsymbol P_j\mathfrak P+V_j=\mathfrak P\quad j=1,2,\dots. \end{equation} | (35) |
Without loss of generality we may assume that $V_j=0$ for all sufficiently large values of $j$ (for minimal normal forms this condition is automatically satisfied).
Note that the choice of an affine normal form is by no means unique: moreover, being a normal form is an open property (small perturbation of the subspaces $V_j$ does not violate the property (35) .
These definitions are tailored to make the following statement trivial.
One possibility to chose an affine normal form is to stick to the polynomials of minimal degree modulo $p_0$. Denote by $w_j\in\C[\eu]$ the greatest common divisor $w_j=\gcd(p_0,p_0\sh j)$; this is a polynomial of degree $\nu_j\le n-1$.
Note that the family of the subspaces $V_j\simeq\C_{\nu_j-1}[\eu]$ is a minimal normal form.
A different strategy of choice of the subspaces $\{V_j\}$ constituting a normal form, is to reproduce the strategy which results in the Poincaré--Dulac normal form for Fuchsian systems with diagonal residue matrix $A\in\Mat(n,\C)$. Recall that in this case instead of solving the homological Eq. (28) , one has to solve matrix equations of the form $[A, H]+jH=B_j$, where $B_j$ are given matrices from $\Mat(n,\C)$, cf. with Ilyashenko and Yakovenko ([Ilyashenko and Yakovenko 2008], Theorem 16.15). The operator taking a matrix $H$ into the twisted commutator as above, is diagonal in the natural basis of matrices having only one nonzero entry, and kernel of this operator is naturally complementary to its image.
An analogous construction can be applied in the case of operators $\boldsymbol P_j$ if the polynomial $p_0=\E(L)$ has simple roots. Then multiplication by any polynomial, including $w_j$, is diagonal, hence one can choose $V_j=\operatorname{Ker}\boldsymbol P_j$. The polynomials $q_j$ which appear in the corresponding normal form, will be vanishing at all roots of $p_0/w_j$, hence divisible by the latter polynomial (recall that we consider polynomials of degree $\le \deg p_0-1$). In particular, if a certain root $\l_i$ of $p_0$ does not appear in any resonance, then all polynomials $q_j$ in the normal form will be divisible by $\eu-\l_i$, and therefore the operator $M$ in the normal form established in Theorem 6 will be divisible (from the right) by the first order Euler operators $\eu-\l_i\in\F$.
This claim gives a partial effective factorization of the normal form (29) , which allows to identify factors with $r_i=0$. In the next section we explain how one can give an accurate description of the factors in (29) in general.
Occurrence of resonances between the roots $\L=\{\l_1,\dots,\l_n\}\subset \C$ of the polynomial $p_0\in\C_n[\eu]$ allows to introduce certain combinatorial structures. First, the (natural linear) order on $\Z$ induces a partial order on the roots: $\l_i\ge \l_k \iff\l_i-\l_k\in\Z_+$.
Second, for each order we can list all roots which produce resonances of this order. Given a natural index $j\in\N$, we define
\begin{equation}\label{equation.5.36} \L_j=\{\l\in\L\colon \l+j\in\L\}\subset\L,\quad j=1,2,\dots. \end{equation} | (36) |
This definition, unambiguous in the case where $p_0$ has only simple roots, should be modified as follows: if $\mu+j=\varkappa$ are two roots in resonance and the multiplicities of $\mu,\varkappa$ in $\L$ (the list which now may have repetitions) are $m,k$ respectively, then $\mu$ enters $\L_j$ with the multiplicity equal to $\min(m,k)$, that is, with its multiplicity as the root of the polynomial $w_j=\gcd (p_0,p_0\sh {j})$.
Together with the sets $\L_j\subset\L$ it is convenient to consider
also their fully ordered counterparts (cf. with
Remark 10), the sets of the corresponding indices
$I_j\subset\{1,2,\dots,n\}$. In the case where $\l$ a root of $p_0$
with multiplicity $m>1$ and of $p_0\sh j$ with multiplicity $k
The dual description can be given by the sets $J(\l)$, which for any root $\l\in\L$ consists of the natural numbers $j\in\N$ such that $\l+j\in\L$. The case of multiple roots needs no special treatment.
Recall that support (or the Newton diagram) of a polynomial $r=\sum c_k t^k\in\C[t]$ is the set of indices $k\in\N$ such that the corresponding coefficient $c_k$ is nonzero: $\supp r=\{k:c_k\ne 0\}\subset\N$.
\begin{align}\label{equation.5.37} L&=(\eu-\l_1+r_1(t))\cdots(\eu-\l_n+r_n(t)),\nonumber \\ r_i&\in\C[t],\quad \supp r_i\subseteq J(\l_i),\quad i=1,\dots,n. \end{align} | (37) |
The rest of this section contains the proof of this theorem.
From that moment we assume that the roots $\l_i$ are labeled in a natural order, see Remark 10.
Consider the operators $E_{ij}\in\F$ of the form (37) in the case where only one polynomial $r_i$ is different from zero and is itself a monomial of degree $j$: $$ E_{ij}=(\eu-\l_1)\cdots(\eu-\l_{i-1})(\eu-\l_i+t^j)(\eu-\l_{i+1})\cdots(\eu-\l_n),\quad i=1,\dots,n. $$ After complete expansion of $E_{ij}$ we obtain
\begin{align}\label{equation.5.38} E_{ij}&=p_0(\eu)+t^j p_{ij}(\eu),\quad p_{ij}\in\C[\eu],\quad i=1,\dots,n, \nonumber \\ p_{ij}&=(\eu-\l_1+j)\cdots(\eu-\l_{i-1}+j)(\eu-\l_{i+1})\cdots(\eu-\l_n). \end{align} | (38) |
A minor modification of this argument proves a similar statement.
Note that the polynomials $p_{ij}$ and $p_{i+1,j}$ in general are different even if $\l_i=\l_{i+1}$.
This means that in the ordered subsequence $\frac{p_{ij}}{w_j}$, $i\in I_j$, the behavior of the poles will be as before (either a new pole appears or the order of the previous pole is increased). In both cases the linear dependence is impossible. $\square$
Assume (by way of induction) that the a Fuchsian operator $L\in\F$ is already shown to be $\F$-equivalent to an operator $L_{j-1}\in\F$ whose $(j-1)$-jet is as in (37) , i.e., \begin{align*} L_{j-1}&=(\eu-\l_1+r_{1,j-1})\cdots(\eu-\l_n+r_{n,j-1})+t^jv_j(\eu)+\cdots, \\ r_{i,j-1}&\in\C[t],\quad \supp r_i\subseteq J(\l_i)\cap[1,j-1],\quad i=1,\dots,n. \end{align*} We will show that there exists an operator $L_j$ of the same form but with $\supp r_{i,j}\in J(\l_i)\cap [1,j]$, which is $\F$-equivalent to $L_{j-1}$. Indeed, adding monomials of order $j$ to the polynomials $r_{i,j-1}$, $$ r_{i,j}=r_{i,j-1}+c_it^j, \quad c_i\ne 0\iff j\in J(\l_i),\quad i=1,\dots,n $$ will affect only terms of order $j$ and higher after the expansion: the (polynomial) coefficient $v_j$ will be replaced by $v_j+\sum c_ip_{ij}$ by definition (38) of the polynomials $p_{ij}$. By a suitable choice of the coefficients $c_i$ for $i\in I_j$, one can bring this sum into the range of the homological operator $\boldsymbol{P}_{j}$, as follows from Corollary 3.
For this choice the homological Eq. (28) will be solvable with respect to $h_j,k_j$ by setting $h_0=1$, $q_j=-\sum c_i p_{ij}$. Continuing this way, we eventually reach the values of $j$ which exceed the maximal order $N$ of possible resonances. The corresponding operator $L_N$, by construction $\F$-equivalent to the initial operator $L$, is $\F$-equivalent to its product part $\prod_{i=1}^n(\eu-\l_i+r_{iN}(t))$ with $\supp r_{iN}\in J(\l_i)$ by Proposition 3.
The minimality of the normal form (37) does not imply that coefficients of the first order factors $r_1,\dots,r_n\in\C[t]$ are $\F$-invariant. Nevertheless, one can expect that for operators of sufficiently high order there will appear moduli (numeric invariants) of $\F$-classification: for holomorphic gauge classification of Fuchsian systems this was discovered by [Kleptsyn and Rabinovich 2004].
Here we prove that the formal and analytic Fuchsian classifications for Fuchsian operators coincide.
More precisely, assume that two formal operators $H,K\in\^\F$, $H=\sum_{k=0}^{n-1}u_k(t)\eu^k$, $K=\sum_{k=0}^{n-1}v_k(t)\eu^k$ with formal coefficients $u_k,v_k\in\C[[t]]$ conjugate two Fuchsian operators $L=\sum_{k=0}^n a_k(t)\eu^k$, $M=\sum_{k=0}^n b_k(t)\eu^k$ with analytic coefficients $a_k, b_k\in\Ox(\C,0)$, $a_n(0)b_n(0)\ne0$.
\begin{equation}\label{equation.6.39} \biggl(\sum_{k=0}^n b_k(t)\eu^k\biggr)\biggl(\sum_{k=0}^{n-1}u_k(t)\eu^k\biggr)= \biggl(\sum_{k=0}^{n-1}v_k(t)\eu^k\biggr)\biggl(\sum_{k=0}^n a_k(t)\eu^k\biggr). \end{equation} | (39) |
We claim that this identity implies that the coefficients $u_k(t)$ of the operator $H$, after passing to a companion form (20) , together satisfy a Fuchsian system of linear ordinary differential equation. This follows from the direct inspection of the way the highest order derivatives of $u_k$ enter the expressions in (39) .
The identity (39) , using the commutation relationship in the Weyl algebra
\begin{equation}\label{equation.6.40} \eu f=f\eu+ g,\quad g=\eu(f)\in\Ox(\C,0)\text{ the Euler derivative of $f$}, \end{equation} | (40) |
One can instantly verify that these equations have the following structure.
The coefficients with which the variables $v_k$ enter the linear forms $r_j$, form an ``upper triangular'' $n\times 2n$-matrix with the same invertible diagonal entry $a_n$: the highest number forms $r_{2n-1},\dots,r_{2n-k}$ depend only on the variables $v_{n-1},\dots,v_{n-k}$, and the variable $v_{n-k}$ enters with the coefficient $a_n$ for all $k=1,\dots,n$.
The coefficients with which the highest order derivatives $u_{kn}$ enter the linear forms $l_j$, are zero for $l_{2n-1},\dots, l_{n}$ and form a ``diagonal'' $n\times 2n$-matrix with the same invertible diagonal entry $b_n$ in the forms $l_{n-1},\dots,l_0$. Indeed, the formulas (40) imply that a highest order derivative $u_{kn}$ can appear only after iterated transposition with the term $b_n\eu^n$ and only before the powers of the type $\eu^{j-n}$.
Together these two observations imply that the system of the linear equations $l_j=r_j$, $j=2n-1,\dots,1,0$ can be resolved with respect to the variables $u_{kn},v_k$, in particular,
\begin{equation}\label{equation.6.41} u_{kn}(t)=\sum_{i=0}^{n-1}\sum_{j=0}^{n-1}c_{knij}(t)u_{ij}(t), \quad c_{knij}\in\Ox(\C,0),\;k=0,\dots,n-1 \end{equation} | (41) |
This system of $n$ linear ordinary differential equations of order $n$ with respect to the functions $u_k=u_k(t)$ is explicitly resolved with respect to the highest order derivatives, hence is a Fuchsian system of $n^2$ first order equations in exactly the same way as in (20) .
It remains only to refer to the well-known fact: any formal solution of a Fuchsian system converges, see Ilyashenko and Yakovenko ([Ilyashenko and Yakovenko 2008], Lemma 16.17 and Theorem 16.16). Thus any noncommutative series for an operator $H$ conjugating two Fuchsian operators $L,M$, converges. Convergence of the series for $K$ follows by the uniqueness of the right division of $MH$ by $L$. $\square$