论文标题
分级代数与规定的希尔伯特系列
Graded algebras with prescribed Hilbert series
论文作者
论文摘要
对于任何功率系列$ a（t）$，带有指数界的非负整数系数，我们建议简单地构造有限生成的单一单元关联代数$ r $，其Hilbert Series $ h（r，t）$非常接近$ a（t）$。如果$ a（t）$是理性的/代数/超越，则相同的是$ h（r，t）$。如果$ a（t）$的系数的增长是多项式的，那么我们以相同的方式构建了一个分级的代数$ r $，以保留其Hilbert Series $ h（r，t）$的系数的多项式增长。我们从1906年开始采用FATOU的经典结果，我们得到的是，如果有限生成的分级代数$ r $具有有限的Gelfand-Kirillov尺寸，那么其Hilbert系列是理性的，也是先验的。特别是，与多项式身份的Hilbert系列有限产生的代数$ r $相同的二分法也具有。
For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$ is rational/algebraic/transcendental, then the same is $H(R,t)$. If the growth of the coefficients of $a(t)$ is polynomial, in the same way we construct a graded algebra $R$ preserving the polynomial growth of the coefficients of its Hilbert series $H(R,t)$. Applying a classical result of Fatou from 1906 we obtain that if a finitely generated graded algebra $R$ 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 finitely generated algebras $R$ with polynomial identity.
