Abstract
Differential equations have arithmetic analogues (Buium in Arithmetic differential equations, Mathematical Surveys and Monographs, vol 118. American Mathematical Society, Providence 2005) in which derivatives are replaced by Fermat quotients; these analogues are called arithmetic differential equations, and the present paper is concerned with the “linear” ones. The equations themselves were introduced in a previous paper (Buium and Dupuy, in Arithmetic differential equations on \(GL_{n}\), II: arithmetic Lie–Cartan theory, arXiv:1308.0744). In the present paper we deal with the solutions of these equations as well as with the Galois groups attached to the solutions.
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References
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Acknowledgments
The authors are indebted to P. Cartier for inspiring discussions. Also the first author would like to acknowledge partial support from the Hausdorff Institute of Mathematics in Bonn, from the NSF through grant DMS 0852591, from the Simons Foundation (Award 311773), and from the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, Project Number PN-II-ID-PCE-2012-4-0201.
