Conservation of number, difference equations, and a technical problem in positive characteristic.

The number of solutions of a given algebro-geometric configuration, when it is finite, does not change upon a small perturbation of the parameters; this persists 
even upon specializations that change the topology.    The precise formulation of this principle of Poncelet and Schubert   required, i.a., the notions of   algebraically closed fields, flatness, completenesss, multiplicity.     I will explain a model-theoretic version, presented in   quite different terms.  It applies notably to difference equations involving the Galois-Frobenius automorphism $x^p$, uniformly in a prime $p$.   In fixed positive characteristic, interesting technical problems arise that I will discuss if time permits.