Formal reduction of singular linear differential systems using eigenrings: A refined approach

Abstract

This paper provides a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. This allows us to establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple and examples are given in Saade (2018).

Introduction

Throughout this paper, $\mathcal{C}$ denotes an effective subfield of $\mathbb{C}$ such that polynomials with coefficients in $\mathcal{C}$ can effectively be factored. For clarity's sake, we first assume (until section 4) that $\mathcal{C}$ is algebraically closed. We denote by $\mathcal{C}((x))$ the field of formal Laurent series in x with coefficients in $\mathcal{C}$ and consider a first-order linear differential system with coefficients in $\mathcal{C}((x))$ of size n and Poincaré rank $q>0$

$$[A]:\frac{dY}{dx}=A(x)Y.$$

We have $A=\frac{1}{x^{q+1}}{\sum }_{k=0}^{\infty }{A}_k{x}^k$ where the matrices $A_k$ are of size n, they have coefficients in $\mathcal{C}$ and $A_0\ne 0$.

In Balser et al. (1979), Balser, Jurkat and Lutz show that such a system has a formal fundamental solution matrix of the form

$$F(x)G(x)=F(x)\mathrm{diag}(G_1(x),\dots ,G_{\ell }(x))$$

where $F\in G{L}_n(\mathcal{C}((x)))$ and G is a block-diagonal matrix composed of blocks $G_i$, each of the form

$$G_i=x^{{\mathrm{\Lambda }}_i}{U}_i\exp \left(Q_i\right),$$

where

• $Q_i$ is a diagonal matrix of the form

  $$Q_i=q_i(\frac{1}{t})I_{m_i}\oplus q_i(\frac{1}{{\omega }_{i}t})I_{m_i}\oplus \dots \oplus q_i(\frac{1}{{{\omega }_i}^{r_i-1}t})I_{m_i},$$

  where $t=x^{\frac{1}{r_i}}$, with $r_i$ minimal, $q_i$ is a polynomial without constant term, ${\omega }_i$ is a primitive $r_i$-th root of unity and $I_{m_i}$ is the identity matrix;

• ${\mathrm{\Lambda }}_i$ is an upper-triangular constant matrix

  $${\mathrm{\Lambda }}_i=J_{m_i}({\lambda }_i)\oplus (J_{m_i}({\lambda }_i)+(\frac{1}{r_i})I_{m_i})\oplus \dots \oplus (J_{m_i}({\lambda }_i)+(\frac{r_i-1}{r_i})I_{m_i})$$

  where $J_{m_i}({\lambda }_i)$ is an $m_i$-dimensional Jordan block, with an eigenvalue ${\lambda }_i$;

• $U_i=\left({\omega }_i^{(j-1)(k-1)}{I}_{m_i}\right)$ is a generalized Vandermonde matrix.

The $q_i$'s are called the exponential parts of the system $[A]$. Note that $G_i$ is determined by $(q_i,{\lambda }_i,r_i,m_i)$. We call $r_i$ (respectively, $m_i$) the ramification index, (respectively, the multiplicity) of $q_i$ in $G_i$.

In the sequel, we will refer to this result as BJL Theorem. Following the formal classification in Balser et al. (1979), we deduce the following theorem, which is a consequence of Theorem I in Balser et al. (1979).

Theorem 1

For a system $[A]$ with $A=\frac{1}{x^{q+1}}{\sum }_{k=0}^{\infty }{A}_k{x}^k$, there exists $P\in G{L}_n(\mathcal{C}((x)))$ such that the change of variable $Y=PZ$ reduces system $[A]$ to an equivalent block-diagonal system

$$[A_1(x)]\oplus \dots \oplus [A_{\ell }(x)]$$

where each block $[A_i]$ is indecomposable over $\mathcal{C}((x))$. Moreover, for each i, there exists a permutation matrix $P_i$ such that

$$P_i[A_i]:=P_i^{-1}{A}_i{P}_i-P_i^{-1}\frac{d{P}_i}{dx}=\left(\begin{array}{ccc}\hfill B_i\hfill & \hfill \frac{1}{x}{I}_{r_i}\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \ddots \hfill & \hfill \frac{1}{x}{I}_{r_i}\hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill B_i\hfill \end{array}\right),$$

where $[B_i]$ is irreducible over $\mathcal{C}((x))$ of size $r_i$ and is repeated $m_i$ times. Each $[B_i]$ is decomposable in $\mathcal{C}((x^{1/r_i}))$. More precisely, let $t:=x^{1/r_i}$ and let $[{\mathcal{B}}_i]$ denote the resulting system over $\mathcal{C}((t))$

$$[{\mathcal{B}}_i]:\frac{dY}{dt}={\mathcal{B}}_{i}Y,{\mathcal{B}}_i:=r_i{t}^{r_i-1}{B}_i(t^{r_i}).$$

Then there exists a transformation $S_i$ in $G{L}_{r_i}(\mathcal{C}((t)))$ such that $S_i[{\mathcal{B}}_i]:=S_i^{-1}{\mathcal{B}}_i{S}_i-S_i^{-1}\frac{d{S}_i}{dt}$ is a diagonal matrix of the form:

$$S_i[{\mathcal{B}}_i]=\mathrm{diag}(W_i(t),\dots ,W_i({{\omega }_i}^{r_i-1}t)),$$

where $W_i(t)=\frac{d{q}_i(t)}{dt}+\frac{{\tilde{\lambda }}_i}{r_i{t}^r}$ and ${\tilde{\lambda }}_i$ differ from ${\lambda }_i$ in (5) by an integer.

This theorem guarantees the existence of transformations that will take the system $[A]$ into the normal form described above from which a fundamental matrix of formal solutions is straightforwardly derived. Formal reduction is the process of finding such transformations. In the present paper, we develop an algorithm which constructs such transformations. Our approach is to revisit the main steps of Theorem 1.

• Using a transformation with coefficients in $\mathcal{C}((x))$ computed from the eigenring of $[A]$, we perform a first maximal decomposition. This decomposition provides a block-diagonal system as in (6), where each subsystem is indecomposable and has only one type of exponential part (Proposition 7).

• Using the structure of each indecomposable subsystem (Proposition 6 and Proposition 7), we obtain the multiplicity of the corresponding exponential part and its ramification (Proposition 8).

• With a transformation in $\mathcal{C}((x))$, we obtain the irreducible system $[B_i]$ of (7) (Proposition 10).

• After applying the appropriate ramification to the system $[B_i]$, we can compute successively the coefficients of its exponential part using at each step the Moser-reduction algorithm Moser (1959/1960) or the splitting lemma as in Barkatou (1997).

The Moser-reduction algorithm and the splitting lemma (see section 2) are two important tools in the formal reduction process. The former produces, for a given system, an equivalent one with minimal Poincaré rank; the latter allows to decouple a given system with a leading matrix $A_0$ having two or more eigenvalues, into subsystems of smaller size each having a leading matrix with a single eigenvalue. The leading coefficients of the exponential parts are given by the nonzero eigenvalues. If the leading matrix is nilpotent and the Poincaré rank is minimal then it is necessary Barkatou (1997) to introduce a ramification ($t=x^{1/s}$) in order to be able to decouple the system again. Various strategies were developed (see for example in Barkatou (1997); Barkatou and Pflügel (1999); Wasow (1965)) to find a suitable integer s and compute transformations (in the new variable $t=x^{1/s}$) in order to obtain a system having a leading matrix with several eigenvalues. In Barkatou (1997), Barkatou proves that it is sufficient to take s as the denominator of the so-called Katz invariant (the “degree” in $x^{-1}$ of the exponential part) of the system which can be computed by an algorithm described in the same paper. The algorithm in Barkatou (1997) for computing the exponential parts, suffers from a practical drawback, namely it could introduce unnecessary ramifications during the formal reduction process; this may happen in the case when the input system has exponential parts with different ramification indices. With our new approach, we avoid this kind of drawback by first splitting the system into indecomposable subsystems with different exponential parts for which we are able to compute the minimal ramification index. Furthermore, once we perform this minimal ramification, we know that no further ramification is needed, so that we can proceed as in (Barkatou, 1997) (using only the splitting lemma and Moser-reduction) to find the corresponding exponential parts of each subsystem.

This paper is an expanded version of our conference paper Barkatou et al. (2018). The additions are as follows. First, a combination of our algorithm and of Barkatou's in (Barkatou, 1997) provides a more efficient algorithm (Algorithm 2) by avoiding some artificial algebraic extensions that could appear during the computation even though they do not appear in the result; this occurs in cases when the same exponential part occurs more than once in a direct sum (see Example 12 and Lemma 14). We present this combination in subsection 5.1.

We observe that, when performing a decomposition using the eigenring, each resulting subsystem has exactly one exponential part, modulo conjugation. It follows that each subsystem has only one slope in the Newton polygon at each step of a reduction process (for more details about the definition of the Newton polygon of a system, see Barkatou (2018)); this gives us the Katz invariant using an explicit formula (see Remark 6 in Barkatou (2018)). Thus we may reinvestigate, see section 5.2, the “smart ramification” approach of Barkatou from (Barkatou, 1997) to minimize the algebraic extensions required to express solutions.

We provide proofs of genericity results, for example Lemma 14 and Proposition 15 using Appendix A, which had been omitted for lack of space. We also introduce a heuristic, see section 5.3, which computes faster a part of the eigenring and which accelerates notably the implementation in many situations. Throughout the paper, detailed examples are proposed to explain and justify the contributions; these and more examples can be found in Saade (2018).

Finally, as we work with Laurent series, we will establish estimates for the number of terms of the series which are required for three basic operations in our setting: computing the characteristic polynomial of a Laurent series matrix in the eigenring, computing a basis of kernels of matrices and computing enough terms in an equivalent system to guarantee that we finally find the exponential parts.