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Newly reducible polynomial iterates
International Journal of Number Theory ( IF 0.7 ) Pub Date : 2021-02-05 , DOI: 10.1142/s1793042121500433 Peter Illig 1 , Rafe Jones 1 , Eli Orvis 2 , Yukihiko Segawa 1 , Nick Spinale 1
International Journal of Number Theory ( IF 0.7 ) Pub Date : 2021-02-05 , DOI: 10.1142/s1793042121500433 Peter Illig 1 , Rafe Jones 1 , Eli Orvis 2 , Yukihiko Segawa 1 , Nick Spinale 1
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Given a field K and n > 1 , we say that a polynomial f ∈ K [ x ] has newly reducible n th iterate over K if f n − 1 is irreducible over K , but f n is not (here f i denotes the i th iterate of f ). We pose the problem of characterizing, for given d , n > 1 , fields K such that there exists f ∈ K [ x ] of degree d with newly reducible n th iterate, and the similar problem for fields admitting infinitely many such f . We give results in the cases ( d , n ) ∈ { ( 2 , 3 ) , ( 3 , 2 ) , ( 4 , 2 ) } as well as for ( d , 2 ) when d ≡ 2 mod 4 . In particular, we show that for all these ( d , n ) pairs, there are infinitely many monic f ∈ ℤ [ x ] of degree d with newly reducible n th iterate over ℚ . Curiously, the minimal polynomial x 2 − x − 1 of the golden ratio is one example of f ∈ ℤ [ x ] with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.
中文翻译:
新可约多项式迭代
给定一个字段ķ 和n > 1 , 我们说一个多项式F ∈ ķ [ X ] 有新约n 迭代ķ 如果F n - 1 是不可约的ķ , 但F n 不在这里F 一世 表示一世 的迭代F )。我们提出表征问题,因为给定d , n > 1 , 字段ķ 使得存在F ∈ ķ [ X ] 学位d 与新约n 迭代,并且对于允许无限多这样的字段的类似问题F . 我们在案例中给出结果( d , n ) ∈ { ( 2 , 3 ) , ( 3 , 2 ) , ( 4 , 2 ) } 以及对于( d , 2 ) 什么时候d ≡ 2 模组 4 . 特别是,我们表明对于所有这些( d , n ) 对,有无限多的monicF ∈ ℤ [ X ] 学位d 与新约n 迭代ℚ . 奇怪的是,最小多项式X 2 - X - 1 黄金比例就是一个例子F ∈ ℤ [ X ] 具有新可约的第三次迭代;很少有其他示例具有较小的系数。我们的调查引发了一些猜想和悬而未决的问题。
更新日期:2021-02-05
中文翻译:
新可约多项式迭代
给定一个字段