论文标题
分级代数，代数功能，平面树和椭圆形积分
Graded Algebras, Algebraic Functions, Planar Trees, and Elliptic Integrals
论文作者
论文摘要
本文调查导致分级代数及其希尔伯特系列。我们提供了有限生成的分级关联代数$ r $的简单构造，而Hilbert系列$ H（R，T）$非常接近任意功率系列$ A（T）$，具有指数限制的非负整数系数。然后，我们总结了具有多项式身份的代数上的一些相关事实。此外，我们讨论了如何找到$ {\ mathbb q}（t）$的Rational/代数/超越性的系列$ A（T）$。使用法图的经典结果，我们得出结论，如果有限生成的分级代数具有有限的gelfand-kirillov尺寸，则其希尔伯特系列是理性的，要么是先验。特别是，具有多项式认同的有限产生的代数，相同的二分法也适用于希尔伯特系列。我们展示了如何使用平面生根的树来产生代数功率系列。最后，我们调查了一些关于非交通性不变理论的结果，这些理论表明我们可以作为希尔伯特系列的各种代数函数甚至椭圆形积分获得。
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.
