论文标题
旨在概括范德沃登（Van der Waerden
Towards a generalization of the van der Waerden's conjecture for Sn-polynomials with integral coefficients over a fixed number field extension
论文作者
论文摘要
Van der Waerden的猜想指出，Monic Integer polyenomials $ f（x）$ $ n $的$ \ Mathscr {p} _ {n，n，n，n}^0（\ Mathbb {q}）$ $ n $ $ n $的$ f（x）$ n $，带有高度$ \ le n $，以使galois $ g_ g_ math of galois $ g_ { $ k_f/\ mathbb {q} $是完整的对称组，具有订单$ | \ Mathscr {p} _ {n，n，n}^0（\ Mathbb {q}）| =（2n）^n+o_n（n+o_n（n^n-1}）$ as $ n \ rightarrow+firtarrow+\ fifty $ \ fyfty $。 Van der Waerden，Chow和Dietmann先前已经针对立方和四分之一的多项式显示了猜想。随后，巴尔加瓦以$ n \ ge6 $证明了这一点。在本文中，我们概括了在代数整数中具有系数的多项式的结果，该固定有限扩展名的$ k/\ mathbb {q} $ $ d $的固定有限扩展名的$ \ mathcal {o} _k $，对于某些$ n $ n $和$ d $。
The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field $K_f/\mathbb{Q}$ is the full symmetric group, has order $|\mathscr{P}_{n,N}^0(\mathbb{Q})|=(2N)^n+O_n(N^{n-1})$ as $N\rightarrow+\infty$. The conjecture has been shown previously for cubic and quartics polynomials by van der Waerden, Chow and Dietmann. Subsequently, Bhargava proved it for $n\ge6$. In this paper, we generalize the result for polynomials with coefficients in the ring of algebraic integers $\mathcal{O}_K$ of a fixed finite extension $K/\mathbb{Q}$ of degree $d$, for some values of $n$ and $d$.
