Geometric constructions in the theory of analytic complexity

Аннотация

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.