The Casoratian and \(p\)-adic difference equations (Q2452085)

The Casoratian (a generalized Casorati determinant) of a system \(f_1,\ldots ,f_m\) of meromorphic functions on a disk in \(\mathbb C_p\) is \[ C(f_1,\ldots ,f_m,n_1,\ldots ,n_m)=\begin{vmatrix} \Delta^{n_1}(f_1) & \ldots & \Delta^{n_m}(f_1)\\ \Delta^{n_1}(f_2) & \ldots & \Delta^{n_m}(f_2)\\ \vdots & \ldots & \vdots\\ \Delta^{n_1}(f_m) & \ldots & \Delta^{n_m}(f_m) \end{vmatrix} \] where \(n_1,\ldots ,n_m\) are natural numbers, \((\Delta (f))(x)=f(x+1)-f(x)\). A study of Casoratians is performed and applied to linear difference equations with coefficients in \(\mathbb C_p[x]\). In particular, the author proves that if such an equation of order \(k\) has \(k\) solutions, linearly independent over \(\mathbb C_p\) and meromorphic on the whole \(\mathbb C_p\), then the equation has \(k\) linearly independent rational solutions. A similar property is found in the ``global'' setting, for difference equations with coefficients from \(\mathbb Q[x]\). Here the existence of systems of entire solutions over \(\mathbb C_p\) corresponding to an infinite set of primes \(p\), implies the existence of linearly independent polynomial solutions with coefficients from \(\mathbb Q\).