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Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals

This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with exponentially bounded nonnegative integer coefficients. Then we summarize some related facts on algebras with polynomial identity. Further we discuss the problem how to find series $a(t)$ which are rational/algebraic/transcendental over ${\mathbb Q}(t)$. Applying a classical result of Fatou we conclude that if a finitely generated graded algebra has a finite Gelfand-Kirillov dimension, then its Hilbert series is either rational or transcendental. In particular the same dichotomy holds for the Hilbert series of a finitely generated algebra with polynomial identity. We show how to use planar rooted trees to produce algebraic power series. Finally we survey some results on noncommutative invariant theory which show that we can obtain as Hilbert series various algebraic functions and even elliptic integrals.