论文标题
多项式图的还原纤维
Reducible Fibers of Polynomial Maps
论文作者
论文摘要
对于$ n $ polyenmial $ f $而言，纤维$ f^{ - 1}（a）中的元素是$ n $ of $ \ mathbb q $的$ n $，对于大多数理性值$ a $ a By Hilbert's nordredusible the rodredubility theorem。在没有此属性的情况下，确定一组出色的$ a $是一个长期存在的开放问题，与Davenport - Lewis--schinzel问题（1959）有关分离多项式的降低性。与以前主要涉及不可分解的$ f $的工作相反，我们回答了可分解$ f = f_1 \ circ \ cdots \ circ f_r $的两个问题，只要不可分解的因素$ f_i \ in \ mathbb q [x] $至少是$ 5 $ $ x^n $或a polyn linial linials linials linours linours linours。
For a degree $n$ polynomial $f$ over the rationals, the elements in the fiber $f^{-1}(a)$ are of degree $n$ over $\mathbb Q$ for most rational values $a$ by Hilbert's irreducibility theorem. Determining the set of exceptional $a$'s without this property is a long standing open problem that is closely related to the Davenport--Lewis--Schinzel problem (1959) on reducibility of separated polynomials. As opposed to previous work which mostly concerns indecomposable $f$, we answer both problems for decomposable $f=f_1\circ\cdots\circ f_r$, as long as the indecomposable factors $f_i\in\mathbb Q[x]$ are of degree at least $5$ and are not $x^n$ or a Chebyshev polynomial composed with linear polynomials.
