论文标题
置换系数取决于参数的单变量多项式的根
Permuting the roots of univariate polynomials whose coefficients depend on parameters
论文作者
论文摘要
我们解决了两个相互关联的问题，这些问题是关于系数取决于参数的单变量多项式根源的置换。首先，我们计算多项式的Galois组$φ（x）\ in \ mathbb {c} [t_1，\ cdots，t_k] [x] $ over $ \ mathbb {c}（t_1，t_1，\ cdots，cdots，t_k）$。前提是相应的多元多项式$φ（x，t_1，\ cdots，t_k）$对于其支持$ a \ subset \ mathbb {z}^{k+1} $，我们确定与任何$ a $相关的galois组。其次，我们确定表格$ p（x，t）= q（t）= 0 $的多项式方程组的Galois组组，其中$ p $和$ q $已固定支持$ a_1 \ subset \ subset \ subbb {z}^2 $和$ a_2 \ a_2 \ subset \ subset \ subset \ subset \} 0 \ \} \ \ \ \ \ \ \ \ \ \ mathbb =对于每个问题，我们确定适当的编织单构图图的图像，以计算所寻求的Galois组。在应用程序中，我们确定了与其支持有关的任何合理函数的Galois组。我们还为代数群体的Galois列举问题组提供了一般性障碍。
We address two interrelated problems concerning permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials $φ(x)\in\mathbb{C}[t_1,\cdots,t_k][x]$ over $\mathbb{C}(t_1,\cdots,t_k)$. Provided that the corresponding multivariate polynomial $φ(x,t_1,\cdots,t_k)$ is generic with respect to its support $A\subset \mathbb{Z}^{k+1}$, we determine the Galois group associated to any $A$. Second, we determine the Galois group of systems of polynomial equations of the form $p(x,t)=q(t)=0$ where $p$ and $q$ have fixed supports $A_1\subset \mathbb{Z}^2$ and $A_2\subset \{0\}\times \mathbb{Z}$, respectively. For each problem, we determine the image of an appropriate braid monodromy map in order to compute the sought Galois group. Among the applications, we determine the Galois group of any rational function that is generic with respect to its support. We also provide general obstructions on the Galois group of enumerative problems over algebraic groups.
