Hermitian matrices depending on three parameters: Coalescing eigenvalues

We consider Hermitian matrix valued functions depending on three parameters that vary in a bounded surface of R3. We study how to detect when such functions have coalescing eigenvalues inside this surface. Our criterion to locate these singularities is based on a construction suggested by Stone [20]. For generic coalescings, any such singularity is related to a particular accumulation of a certain phase, or lack thereof, as we cover the surface.