Refinements of the Gauss-Lucas theorem using rational lemniscates and polar convexity

Abstract:

The classical Gauss-Lucas theorem for complex polynomials $p(z) ≔(z-z_1) \cdots (z-z_n)$ states that the critical points of $p(z)$ are in the convex hull of its zeros. We give two refinements of the Gauss-Lucas theorem, both connected by the notion of polar convexity.

In the first, we describe lemniscatic regions that do not contain non-trivial critical points of any polynomial of the form $(z-z_1) \cdots (z-z_m)(z-z^*_{m+1})\cdots (z-z^*_{n})$, when the parameters $z^*_{m+1},\ldots ,z^*_{n}$ vary freely in a specified set, containing the zeros $z_{m+1},\ldots ,z_{n}$.

In the second, we locate the non-trivial critical points of $p(z)$ in the intersection of $m+1$ polar-convex sets, where $m$ is the number of distinct zeros of $p(z)$. This refinement complements recent ones in Dimitrov [Proc. Amer. Math. Soc. 126 (1998), pp. 2065–2070] and in Curgus and Mascioni [Proc. Amer. Math. Soc. 132 (2004), pp. 2973–2981].