Permuting the roots of univariate polynomials whose coefficients depend on parameters

Abstract: 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 $\varphi(x)\in\mathbb{C}[t_1,\cdots,t_k][x]$ over $\mathbb{C}(t_1,\cdots,t_k)$. Provided that the corresponding multivariate polynomial $\varphi(x,t_1,\cdots,t_k)$ 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.