A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number bn 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 =limn→∞bnn 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 bn, n = 1, 2,…, is transcendental. With the same approach we construct examples of free graded semigroups 〈Y 〉 with the following property. If dn is the number of elements of degree n of 〈Y 〉, then the limit δ =limn→∞dnn exists and is transcendental.

中文翻译：

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结合代数和超越级数的极小变体

如果衡量其余维序列增长的变体的指数严格大于其任何适当子变体的指数，则在特征为 0 的域上的各种关联代数称为极小，即其余维序列的增长速度远快于其适当子变体的维数序列。根据 Giambruno 和 Zaicev 的结果，数bn给定指数的最小变体n是有限的。使用整数的有色（或加权）组合理论的方法，我们证明了极限β =林n→∞bnn存在并且可以表示为方程的正解一种(吨) = 0在哪里一种(吨)是一个明确给定的幂级数。对于具有给定 Gelfand-Kirillov 维数的相对自由的秩代数的最小变体的数量，也获得了类似的结果d. 由漏幂级数的经典结果可知，该数列的母函数bn,n = 1, 2,…, 是超越的。用同样的方法，我们构造了自由分级半群的例子〈是 〉具有以下属性。如果dn是度数的元素数n的〈是 〉, 那么极限δ =林n→∞dnn存在并且是超越的。