Let $f_1, f_2, \ldots, f_n$ be linear polynomials in $\ell$ variables, and $\lambda_1, \lambda_2, \ldots, \lambda_n$ nonzero complex numbers. The product $$ \Phi_\lambda=\Prod_{i=1}^n f_1^{\lambda_i}, $$ defines a (multivalued) function on $\ell$-dimensional complex space, or more precisely, on the complement of a set of hyperplanes. Mathematical physics motivates the question of describing its set of critical points in terms of the input polynomials and parameters $\lambda_i$, and "generically" the critical points are well understood in terms of a certain cohomological vanishing condition. In my talk I will describe how the geometry and combinatorics of hyperplane arrangements sheds some light on the non-generic case, by means of a suitable "universal" construction with somewhat remarkable properties. |