论文标题
驯服属0（或1）组件在变量分离方程式上
Taming Genus 0 (or 1) components on variables-separated equations
论文作者
论文摘要
为了查明表单$ c_ {f，g} = {（x，y）|的曲线的属性f（x）-g（y）= 0} $您必须解决其投影标准化的属0和1个组件$ \ tilde c_ {f，g} $。对于$ f $和$ g $多项式，具有$ f $ innecomposable，[fr73a]区分$ \ tilde c_ {f，g} $，$ u = 1 $ vers $ vers $ u> u> u> 1 $组件（schinzel的问题）。对于$ u = 1 $，[prop。 1，fr73b]给出了直接的属公式。要完成$ u> 1 $需要一个临时属计算。 [pak22]放弃了不可兼容和多项式限制，但添加了$ \ tilde c_ {f，g} $是不可约的（$ u = 1 $）。他表明 - 对于固定的$ f $ - 除非$ f $的封面封闭为$ f $属于0或1属，否则该属在DEG（$ g $）上线性增长。方法I和方法II扩展[Prop。 1，FR73B}]使用Nielsen类来概括Pakovich的$ u> 1 $的配方。 方法我在$ f $和$ g $上播放$ z $ - line，$ p^1_z $，我们从中计算了光纤产品。 方法II基于此封面的明确计算分支周期，将投影使用$ y $ line，$ p^1_y $。 Hurwitz家庭追踪这些组件的重要性。扩展[Prop。 2，FR73A]显示了如何解决Pakovich的问题。在没有损失的情况下，从具有相同的galois关闭的（$ f^*，g^*$）开始，并为其规范表示纠缠。因此，它们在纤维产品上产生了多个成分。 然后，我们使用来自方法II的$ w $的分支周期来对可能的组件类型，$ w $分类，这些类型$ w $出现在$ \ tilde c_ {f^*，g^*} $上。结果是一种尼尔森类配方，明确地告诉$ g_1 \，$ s，以避免确保$ \ tilde c_ {f*，g*og_1} $的组成属的增长为deg（$ g_1 $）。特别要注意：使用和扩展尼尔森类别和属0问题的解决方案（对不可分解的有理函数的单型组进行了分类）。
To figure properties of a curve of form $C_{f,g} = {(x,y)| f(x) - g(y)= 0}$ you must address the genus 0 and 1 components of its projective normalization $\tilde C_{f,g}$. For $f$ and $g$ polynomials with $f$ indecomposable, [Fr73a] distinguished $\tilde C_{f,g}$ with $u=1$ versus $u > 1$ components (Schinzel's problem). For $u = 1$, [Prop. 1, Fr73b] gave a direct genus formula. To complete $u > 1$ required an adhoc genus computation. [Pak22] dropped the indecomposable and polynomial restrictions but added $\tilde C_{f,g}$ is irreducible ($u = 1$). He showed - for fixed $f$ - unless the Galois closure of the cover for $f$ has genus 0 or 1, the genus grows linearly in deg($g$). Method I and Method II extend [Prop. 1, Fr73b}] using Nielsen classes to generalize Pakovich's formulation for $u > 1$. Method I plays on the covers $f$ and $g$ to the $z$-line, $P^1_z$, from which we compute the fiber product. Method II uses the projection to the $y$-line, $P^1_y$, based on explicitly computing branch cycles for this cover. Hurwitz families track the significance of these components. Expanding on [Prop. 2, Fr73a] shows how to approach Pakovich's problem. With no loss, start with ($f^*,g^*$) which have the same Galois closures, and for which their canonical representations are entangled. They, therefore, produce more than one component on the fiber product. Then, we classify the possible component types, $W$, that appear on $\tilde C_{f^*,g^*}$ using the branch cycles for $W$ that come from Method II. The result is a Nielsen class formulation telling explicitly what $g_1\,$s to avoid to assure the growth of the component genuses of $\tilde C_{f*,g*og_1}$ as deg($g_1$) increases. Of particular note: using and expanding on Nielsen classes and the solution of the genus 0 problem (classifying the monodromy groups of indecomposable rational functions).
