Discriminant and bifurcation diagram of complex trigonometric polynomials

Abstract

The complex analogues of usual trigonometric polynomials are bipolynomials. The discriminant (resp. bifurcation diagram) is the set of bipolynomials of a given bidegree with less simple roots (resp. distinct critical values) than a generic bipolynomial. The extended discriminant is the union of the discriminant with a given hyperplane. We prove that the complement to the extended discriminant is an Eilenberg-MacLane space K(π, 1) associated to some extension of a Weyl group of series A n . We prove that the cohomology ring of the complement to the discriminant (resp. extended discriminant, bifurcation diagram) stabilises as the bidegree tends to infinity.