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 actionof the gauge group. With the help of the second construction, we obtaina necessary differential algebraic condition for membership of a function in the tangent space to the class of bivariate functionsof analytic complexity $\le 2$ at the point $z_0=x^3 y^2 +xy$. From this result we show that thepolynomial $z=x^3y^2+xy + \pi x^2 y^3$ of degree 5 has analytic complexity 3.

analytic function, analytic complexity, metric space