The singularities of type \(A_ k\) of holonomic systems (Q1331258)

The author considers a microdifferential holonomic system \(M\) whose characteristic variety is in a generic position of the point \((0,dx_ 0) \in P^*X\), \(X = \mathbb{C}^{n+1} \ni (x_ 0, x_ 1, \dots, x_ n)\). The system \(M\) admits a basis \(u_ 1, \dots, u_ m\), satisfying the equations \[ x_ 0u = \bigl( A_ 0(x) + A_ 1 \xi_ 0^{-1} \bigr) u, \quad \xi_ j \xi_ 0^{-1} u = B_ j(x)u, \quad j = 1, \dots, n, \] where \(A_ 0(x)\), \(B_ j(x)\), \(j = 1, \dots, n\), are analytic matrices of order \(m\) and \(A_ 1\) is a constant matrix. The matrices \(A_ 0\), \(B_ j\), \(j = 1, n\), verify some integrability conditions which permit to establish a generating matrix \(H(x)\), such that \[ \left[ \bigl( A_ 1,H(x)\bigr) + H(x), {\partial H \over \partial x_ i} \right] =0, \quad i=1,n,\left[ {\partial H \over \partial x_ j}, {\partial H \over \partial x_ j} \right] = 0, \quad i,j = l,n. \tag{1} \] A matrix \(H(x)\) that satisfies the conditions (1) is called a \(D\)-matrix. Thus, the investigation of holonomic systems reduces to that of \(D\)-matrices. A \(D\)-matrix \(H(x)\), \(x = (x_ 1, \dots, x_ n)\) of order \(n+1\) is of type \(A_{n+1}\) if \(H(x)\) has the properties: \(H(0) = 0\), \({\partial H \over \partial x_ 1} (0) = X\), \({\partial^ 2H \over \partial x^ 2} (0) = Y^ n\), where \(X = \left[ \begin{smallmatrix} 0 & 1 & 0 \\ & & 1 \\ 0 & & 0 \end{smallmatrix} \right]\), \(Y = \left[ \begin{smallmatrix} 0 & & 0 \\ 1 \\ 0 & 1 & 0 \end{smallmatrix} \right]\). In this paper the author proves the following theorem. There is a unique \(D\)-matrix of type \(A_{n+1}\). It follows that the matrix \(A_ 1\) has the form \(A_ 1 = \text{diag} (\;)\).