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The Casoratian and \(p\)-adic difference equations. (Casoratien et équations aux différences \(p\)-adiques.) (French. English summary) Zbl 1314.12001

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 \(\mathbb Q\).

MSC:

12H99 Differential and difference algebra
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
30G06 Non-Archimedean function theory
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[1] Amice (Y.).— Les nombres \(p\)-adiques, Presses universitaires de France, collection SUP (1975). · Zbl 0313.12104
[2] André (Y.).— G-functions and Geometry. Vieweg & Sohn, Braunschweig (1989). · Zbl 0688.10032
[3] André (Y.).— Sur la conjecture des \(p\)-courbures de Grothendieck-Katz et un problème de Dwork. In Geometric aspects of Dwork theory Vol I, II, p. 55-112. Walter de Gruyter Gmbh & co, KG, Berlin (2004). · Zbl 1102.12004
[4] Bézivin (J.-P.).— Wronskien et équations différentielles \(p\)-adiques, Acta Arithmetica 158, p. 61-78 (2013). · Zbl 1278.12004
[5] Barsky (D.), Bézivin (J.-P.).— Article en préparation.
[6] Casorati (P.).— Il calcolo delle differenze finite interpretatoed accresciuto di nuovi teoremi a sussidio pricimalmente delle odierne ricerche basate sulla variabilitá complessa. Annali di Matematica Pura ed Applicata, Series 2, 10, (1), p. 10-45 (1880).
[7] Dwork (B.), Gerotto (G.), Sullivan (F.J.).— An introduction to G-functions. Annals of Math studies, 133, Princeton University Press (1994). · Zbl 0830.12004
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