Geometric constructions in the theory of analytic complexity

Two geometric constructions are considered in the context of analytic complexity. Using the first construction, on the set of analytic functions, we build a metric invariant under the action of the gauge group. With the help of the second construction, we obtain a necessary differential algebraic condition for membership of a function in the tangent space to the class of bivariate functions of analytic complexity $\le 2$ at the point $z_0=x^3 y^2 +xy$. From this result we show that the polynomial $z=x^3y^2+xy + \pi x^2 y^3$ of degree 5 has analytic complexity 3.