Geodesic
Geometric constructions in the theory of analytic complexity
Izvestiya. Mathematics , Tome 88 (2024) no. 3, pp. 411-418

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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.
Keywords: analytic function, analytic complexity, metric space. 
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     title = {Geometric constructions in the theory of analytic complexity},
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V. K. Beloshapka. Geometric constructions in the theory of analytic complexity. Izvestiya. Mathematics , Tome 88 (2024) no. 3, pp. 411-418. http://geodesic.mathdoc.fr/item/IM2_2024_88_3_a0/