论文标题
多项式判别剂的无平方值II
Squarefree values of polynomial discriminants II
论文作者
论文摘要
我们确定具有无平方判别的给定程度的积分二进制形式的密度，首次证明较低的密度为正。此外，我们确定了整数二进制形式的密度，这些二进制形式在数字字段中降低了最大顺序。后者尤其证明了Poonen以$ \ mathbb {p}^1_ \ Mathbb {z} $猜想的``算术bertini定理''。 我们的方法还允许我们证明有$ \ gg x^{1/2+1/（n-1）} $ galois group〜 $ s_n $的学位〜$ n $字段〜$ s_n $，而绝对歧视剂小于$ x $，改善了$ \ gg gg x^^x^^^^^{1/2+1/n} $。 最后，我们的方法纠正了中川的错误，从而在下限上恢复了较早的（缩回）结果，因为完全不受影响的$ a_n $ extensions的数量是有界歧视的二次数字字段。
We determine the density of integral binary forms of given degree that have squarefree discriminant, proving for the first time that the lower density is positive. Furthermore, we determine the density of integral binary forms that cut out maximal orders in number fields. The latter proves, in particular, an ``arithmetic Bertini theorem'' conjectured by Poonen for $\mathbb{P}^1_\mathbb{Z}$. Our methods also allow us to prove that there are $\gg X^{1/2+1/(n-1)}$ number fields of degree~$n$ having associated Galois group~$S_n$ and absolute discriminant less than $X$, improving the best previously known lower bound of $\gg X^{1/2+1/n}$. Finally, our methods correct an error in and thus resurrect earlier (retracted) results of Nakagawa on lower bounds for the number of totally unramified $A_n$-extensions of quadratic number fields of bounded discriminant.
