p^2度的合成和碰撞
Compositions and collisions at degree p^2
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）。为了计算可分解的次数，我们需要知道在适当的归一化下，形式为f = g o h = g^* o h^* 且（g，h）=（g^*，h^*）和deg g = deg g^* 的等度碰撞的次数。这样的碰撞只发生在野外情况下，其中场特性p除以deg f。有限域上可分解数的合理界限是已知的，但在一般情况下，特别是在度为p^2的情况下，界限就不那么明显了。我们给出了所有具有碰撞的p^2次多项式的分类。它给出了特征p的有限域上p^2次可分解多项式的确切个数。我们还提出了一个有效的算法，确定给定的p^2次多项式是否有冲突。
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. In order to count the decomposables, one wants to know, under a suitable normalization, the number of equal-degree collisions of the form f = g o h = g^* o h^* with (g, h) = (g^*, h^*) and deg g = deg g^*. Such collisions only occur in the wild case, where the field characteristic p divides deg f. Reasonable bounds on the number of decomposables over a finite field are known, but they are less sharp in the wild case, in particular for degree p^2. We provide a classification of all polynomials of degree p^2 with a collision. It yields the exact number of decomposable polynomials of degree p^2 over a finite field of characteristic p. We also present an efficient algorithm that determines whether a given polynomial of degree p^2 has a collision or not.
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