关于单参数的参数多项式和一般多项式
On parametric and generic polynomials with one parameter
摘要：
给定域$k \subseteq L$，我们的结果涉及$k$上的单参数$L$ -参数多项式，以及它们与一般多项式的关系.前者是群$G$的多项式$P(T,Y) \in k[T][Y]$，其通过$T$在$L$中的特殊化来参数化群$G$的$L$的所有伽罗瓦扩张，而后者是对于每个域$L \supseteq k$都是$L$参数的多项式。我们表明，例如，是$L$参数与$L$采取的是单字段$\mathbb{C}((V))(U)$实际上是足够的多项式$P(T, Y) \in \mathbb{C}[T][Y]$是通用的。作为推论，我们得到一个完整的列表的一个参数一般多项式在一个给定的领域的特征0，补充经典文献的主题。我们的方法也适用于Schinzel的一个老问题：受桦树和Swinnerton-Dyer猜想，我们提供了一个参数家庭的仿射曲线在数域上，所有的有理点，但没有合理的通用点。
英文摘要：
Given fields $k \subseteq L$, our results concern one parameter $L$-parametric polynomials over $k$, and their relation to generic polynomials. The former are polynomials $P(T,Y) \in k[T][Y]$ of group $G$ which parametrize all Galois extensions of $L$ of group $G$ via specialization of $T$ in $L$, and the latter are those which are $L$-parametric for every field $L \supseteq k$. We show, for example, that being $L$-parametric with $L$ taken to be the single field $\mathbb{C}((V))(U)$ is in fact sufficient for a polynomial $P(T, Y) \in \mathbb{C}[T][Y]$ to be generic. As a corollary, we obtain a complete list of one parameter generic polynomials over a given field of characteristic 0, complementing the classical literature on the topic. Our approach also applies to an old problem of Schinzel: subject to the Birch and Swinnerton-Dyer conjecture, we provide one parameter families of affine curves over number fields, all with a rational point, but with no rational generic point.
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