Analytic Complexity: Gauge Pseudogroup, Its Orbits, and Differential Invariants

All characteristics of analytic complexity of functions are invariant under a certain natural action (gauge pseudogroup $\mathcal G$). For the action of the pseudogroup $\mathcal G$, differential invariants are constructed and the equivalence problem is solved. Functions of two as well as of a greater number of variables are considered. Questions for further analysis are posed.