Loci of complex polynomials, part I

Abstract:

The classical Grace theorem states that every circular domain in the complex plane $\mathbb {C}$ containing the zeros of a polynomial $p(z)$ contains a zero of any of its apolar polynomials. We say that a closed domain $\Omega \subseteq \mathbb {C}^*$ is a locus of $p(z)$ if it contains a zero of any of its apolar polynomials and is the smallest such domain with respect to inclusion. In this work we establish several general properties of the loci and show, in particular, that the property of a set being a locus of a polynomial is preserved under a Möbius transformation. We pose the problem of finding a locus inside the smallest disk containing the roots of $p(z)$ and solve it for polynomials of degree $3$. Numerous examples are given.