驯服分解和碰撞

摘要：

域上的一元多项式f是可分解的，如果对非线性多项式g和h，f = g o h = g（h）。直观地清楚，可分解多项式在有限域上的所有多项式中形成少数。Tame的情况下，其中的特征p的Fq不除n = deg f，是相当好理解的，我们有合理的界限上的次数为n的可分解。然而，没有确切的公式是已知的，如果$n$有两个以上的素因子。为了计算可分解的数，人们想要知道在适当的归一化下碰撞的数量，其中本质上不同的（g，h）产生相同的f。在Tame情形下，Ritt第二定理将所有2-碰撞分类。我们引入了一个规范形式的多碰撞分解的任意长度的参数的（非）唯一性的精确描述。我们得到了一个有效的计算公式的确切数量的这样的碰撞在一个有限域上的特征互质p的次数n。这导致了一个算法的确切数量的可分解多项式在一个有限域Fq在驯服的情况下。

英文摘要：

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The tame case, where the characteristic p of Fq does not divide n = deg f, is fairly well-understood, and we have reasonable bounds on the number of decomposables of degree n. Nevertheless, no exact formula is known if $n$ has more than two prime factors. In order to count the decomposables, one wants to know, under a suitable normalization, the number of collisions, where essentially different (g, h) yield the same f. In the tame case, Ritt's Second Theorem classifies all 2-collisions. We introduce a normal form for multi-collisions of decompositions of arbitrary length with exact description of the (non)uniqueness of the parameters. We obtain an efficiently computable formula for the exact number of such collisions at degree n over a finite field of characteristic coprime to p. This leads to an algorithm for the exact number of decomposable polynomials at degree n over a finite field Fq in the tame case.

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