On Topological Aspects of Numerical Range

For a pair of real quadratic forms in \(n\geq 2\) variables, we consider a natural counterpart of the numerical range of a complex square matrix and present several related topological results. The main attention is given to the singularities of the associated mapping from a real projective space into the plane. For odd \(n\geq 3\) and generic pair of real quadratic forms, we prove that the discriminant curve of the aforementioned mapping always contains an odd number of cusp points. More detailed results are obtained for pairs of binary and ternary quadratic forms using the case-by-case analysis of the normal forms of quadratic mappings into the plane given in recent papers of M. Farnik and Z. Jelonek.