A survey of difference algebraic geometry

A systematic study of difference polynomial equations leads to a development of difference algebraic geometry. Inspired by questions raised by Macintyre regarding the number of solutions of a difference polynomial systems over algebraic closures of finite fields equipped with a power of the Frobenius, and Hrushovski's proof of the twisted Lang-Weil estimate in this context, we pursue some further directions with hope of obtaining finer number-theoretic information about the number of solutions. We will discuss some recent work on a Chebotarev density-style theorem for difference function fields, Galois stratification procedure in difference geometry, and some speculations regarding the etale cohomology of difference schemes.