论文标题
关联代数和先验系列的最小品种
Minimal varieties of associative algebras and transcendental series
论文作者
论文摘要
如果特征0的代数在特征0上的各种代数序列的增长速度比其任何适当的亚变化的编辑序列快得多，则称为最小。根据Giambruno和Zaicev的结果，得出的是，给定指数$ n $的最小品种的数字$ b_n $是有限的。使用整数的有色（或加权）组成理论的方法，我们表明$β= \ lim_ {n \ to \ infty} \ sqrt [n] {b_n} $都存在，并且可以表示为方程式$ a（t）= 0 $ a（t）$ a（t）$ a（t）$ a（t）的积极解决方案。对于给定的Gelfand-Kirillov尺寸，其相对自由的代数为$ d $的最小品种的数量也获得了类似的结果。从脱离功率系列的经典结果来看，序列$ b_n $，$ n = 1,2，\ ldots $的生成函数是先验的。通过相同的方法，我们构建了免费等级的半群的示例$ \ langle y \ rangle $以及以下属性。如果$ d_n $是$ \ langle y \ rangle $的度量$ n $的元素的数量，则限制$δ= \ lim_ {n \ to \ infty} \ sqrt [n] {n] {d_n} $ exists，并且是超定义。
A variety of associative algebras over a field of characteristic 0 is called minimal if its codimension sequence grows much faster than the codimension sequence of any of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number $b_n$ of minimal varieties of given exponent $n$ is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit $β=\lim_{n\to\infty}\sqrt[n]{b_n}$ exists and can be expressed as the positive solution of an equation $a(t)=0$ where $a(t)$ is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand-Kirillov dimension of their relatively free algebras of rank $d$. It follows from classical results on lacunary power series that the generating function of the sequence $b_n$, $n=1,2,\ldots$, is transcendental. With the same approach we construct examples of free graded semigroups $\langle Y\rangle$ with the following property. If $d_n$ is the number of elements of degree $n$ of $\langle Y\rangle$, then the limit $δ=\lim_{n\to\infty}\sqrt[n]{d_n}$ exists and is transcendental.
