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Differential Galois theory has played important roles in the the- ory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order non- linear differential equation. We will define for the differential equation the differential Galois group, will study the structure of the group, and will prove the equivalent between the existence of the Liouvillian first integral and the solvability of the corresponding differential Galois group.

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the $p$-adic counterpart of the GKZ hypergeometric system. In the language of dagger spaces introduced by Grosse-Kl\"onne, the $p$-adic GKZ hypergeometric complex is a twisted relative de Rham complex of meromorphic differential forms with logarithmic poles for an affinoid toric dagger space over the dagger unit polydisc. It is a complex of ${\mathcal O}^\dagger$-modules with integrable connections and with Frobenius structures defined on the dagger unit polydisc such that traces of Frobenius on fibers at Techm\"uller points define the hypergeometric function over the finite field introduced by Gelfand and Graev.

No abstract uploaded!

No abstract uploaded!