Differential modules and singular points of \(p\)-adic differential equations. (English) Zbl 0493.12030

Baldassarri, Francesco

Adv. Math. 44, 155-179 (1982).

This paper deals with a system of ordinary linear differential equations: \[ dY/dx = BY \tag{1} \] where \(B\), \(Y\) are \(n\) by \(n\) matrices and the entries of \(B\) are germs of \(p\)-adic meromorphic functions at \(0\). The formal part of Turrittin’s reduction theory for (1) is first reexamined. After replacing \(x\) by \(t^e\), where \(e = n!\) and \(Y(x)\) by \(U(t)Z(t)\), for a suitable invertible matrix of finite-tailed Laurent series \(U(t)\), the matrix \(Z(t)\) satisfies a system of differential equations of the form: \[ t \frac{dZ}{dt} = CZ \tag{2} \] where \(C\) is in a block-diagonal form: \[ C=\left(\begin{matrix} \begin{matrix} C_1 &O\\ O &C_2\end{matrix} && O\\ &\ddots &\\ O && C_m \end{matrix}\right), \qquad C_i=\left(\begin{matrix} P_i(1/t) &&& 0\\ 1&\ddots&&\\ &\ddots&\ddots&\\ 0&&1&P_i(1/t) \end{matrix}\right)\tag{3} \] where for \(i = 1, \ldots, m\), \(P_i(z)\) denotes a polynomial in \(z\). Let us assume that whenever \(P_i(z) - P_j(z)\) is a constant \(\alpha_{ij}\), then this constant is a \(p\)-adic non-Liouville number, by which we mean that: \[ (\mathrm{ord}(m + \alpha_{ij}) = O(\log m) \quad\text{as }m\to\infty \tag{4} \] where \(\mathrm{ord}\) denotes the \(p\)-adic valuation.
The main result states that under this assumption the matrix \(U(t)\) has a non-zero radius of \(p\)-adic convergence. Condition (4) is automatically satisfied if the coefficients of the Laurent series in \(x\) appearing in \(B\) are algebraic numbers.

Reviewer: Francesco Baldassarri (Padova)

MSC:

12H25	\(p\)-adic differential equations
12H20	Abstract differential equations

Keywords:

singular points; Turrittin normal form; system of ordinary linear differential equations; p-adic meromorphic functions

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