When $D$-companion matrix meets incomplete polynomials
Teng Zhang
TL;DR
The paper generalizes the Gauss-Lucas theorem to convex combinations of incomplete polynomials $A_n^\gamma(z)=\sum_{k=1}^n \gamma_k g_k(z)$, where $g_k(z)$ are the incomplete polynomials associated with the zeros $z_j$ of $A_n(z)=\prod_{j=1}^n(z-z_j)$. By employing $D$-companion matrix techniques and majorization theory, it establishes that zeros of $A_n^\gamma$ lie in the closed convex hull $H(z_1,\dots,z_n)$ and proves a majorization relation $\prod_{j=1}^k |w_j|\le\prod_{j=1}^k |z_j|$ for the ordered zeros $w_j$ of $A_n^\gamma$ and $z_j$ of $A_n$, with a corresponding result for singular values in Theorem 4. It then provides a detailed structural localization: all zeros of $A_n^\gamma$ are contained in explicit discs centered at $c_j=(1-\gamma_j)z_j+\gamma_j z_n$ with radius $(n-2)\gamma_j|z_n-z_j|$, and shows that the union of these discs governs zero locations for all convex combinations. The work also clarifies the geometry of zeros under convex combinations of derivatives and symmetric sums, including a positive result that the zero set S of all convex combinations equals $H(z_1,\dots,z_n)$, while highlighting natural limitations via counterexamples. Overall, the paper provides a simple, constructive framework for zero localization and majorization in the context of incomplete polynomials.
Abstract
In this paper, we provide a simple proof of a generalization of the Gauss-Lucas theorem. By using methods of D-companion matrix, we get the majorization relationship between the zeros of convex combinations of incomplete polynomials and an origin polynomial. Moreover, we prove that the set of all zeros of all convex combinations of incomplete polynomials coincides with the closed convex hull of zeros of the original polynomial. The location of zeros of convex combinations of incomplete polynomials is determined.