Simple connectivity of the D-stability domain and D-stability of polynomial families

It is proved that all monic-polynomials of order n with roots lying in some open region on the complex plane forms a simply connected set in the polynomial parameter space. Based on this result, edge theorems for D-stability of general polyhedrons of polynomials and boundary theorems for D-stability of compact sets of polynomials are obtained. Different from previous ones, our edge theorems and boundary theorems do not require the convexity or the connectivity of the set of polynomials. Moreover, our boundary theorem for families of polynomials with dependent coefficients does not require the coefficient dependency relation to be affine