Sector Analogue of the Gauss–Lucas Theorem

Sendov, Blagovest ; Sendov, Hristo

Canadian journal of mathematics, Tome 73 (2021) no. 2, pp. 318-338

Résumé

The classical Gauss–Lucas theorem states that the critical points of a polynomial with complex coefficients are in the convex hull of its zeros. This fundamental theorem follows from the fact that if all the zeros of a polynomial are in a half plane, then the same is true for its critical points. The main result of this work replaces the half plane with a sector as follows.We show that if the coefficients of a monic polynomial $p(z)$ are in the sector {teiψ : ψ∈ [0, φ], t⩾0}, for some $\unicode[STIX]{x1D719}\in [0,\unicode[STIX]{x1D70B})$, and the zeros are not in its interior, then the critical points of $p(z)$ are also not in the interior of that sector.In addition, we give a necessary condition for a polynomial to satisfy the premise of the main result.

Détail

DOI : 10.4153/S0008414X19000609

Mots-clés : Gauss–Lucas theorem, polynomial, zeros and critical points of polynomial, polynomial with coefficients in a sector, interlacing polynomial, non-convex