Permuting the roots of univariate polynomials whose coefficients depend on parameters

We address two interrelated problems concerning the permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials $\varphi(x)\in\mathbb{C}[y_1,\cdots,y_k][x]$ over $\mathbb{C}(y_1,\cdots,y_k)$. Provided that the corresponding multivariate polynomial $\varphi(x,y_1,\ldots,y_k)$ is generic with respect to its support $A\subset \mathbb{Z}^{k+1}$, we determine the associated Galois group for any such $A$. Second, we determine the Galois group of systems of polynomial equations of the form $p(x,y)=q(y)=0$ 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 generic with respect to its support. We also provide general obstructions to the Galois group of enumerative problems over algebraic groups.