论文标题
希尔伯特·辛奈泽尔专业属性
The Hilbert-Schinzel specialization property
论文作者
论文摘要
我们建立了一个著名的希尔伯特不可约性定理的“ Over the Ring”版本。鉴于$ k+n $变量有限的许多多项式，其系数为$ \ mathbb z $，在最后的$ n $变量中具有正学位，我们表明，如果它们在$ \ mathbb z $上不可约束，并且满足必要的“ schinzel条件”，那么第一个$ k $ a $ k $ a $ k $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $这样的方式使剩下的$ n $变量中的多项式保留了$ {\ mathbb z} $的不可约性。来自schinzel假设的schinzel条件是，当专门为$ {\ mathbb z}^k $中的第一个$ k $变量专门化时，多项式的产物不应始终由某些共同的质量数来除外。我们的结果也改善了schinzel假设的“副本”版本：在某些schinzel条件下，副率多项式假定副本值。我们在$ \ mathbb z $以外的许多其他环上证明了我们的结果，例如最后一个的UFD和Dedekind域。
We establish a version "over the ring" of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in $k+n$ variables, with coefficients in $\mathbb Z$, of positive degree in the last $n$ variables, we show that if they are irreducible over $\mathbb Z$ and satisfy a necessary "Schinzel condition", then the first $k$ variables can be specialized in a Zariski-dense subset of ${\mathbb Z}^k$ in such a way that irreducibility over ${\mathbb Z}$ is preserved for the polynomials in the remaining $n$ variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is that, when specializing the first $k$ variables in ${\mathbb Z}^k$, the product of the polynomials should not always be divisible by some common prime number. Our result also improves on a "coprime" version of the Schinzel Hypothesis: under some Schinzel condition, coprime polynomials assume coprime values. We prove our results over many other rings than $\mathbb Z$, e.g. UFDs and Dedekind domains for the last one.
