Leaky roots and stable Gauss-Lucas theorems

ABSTRACT

Let $p:\mathbb{C}\to \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p^{\prime }\left(z\right)=0$, are contained in the convex hull of its roots. In a recent quantitative version, Totik shows that if almost all roots are contained in a bounded convex domain $K\subset \mathbb{C}$, then almost all roots of the derivative $p^{\prime }$ are in an $\epsilon -$neighborhood $K_{\epsilon }$ (in a precise sense). We prove a quantitative version: if a polynomial p has n roots in K and $\lesssim c_{K,\epsilon }\left(n/\log n\right)$ roots outside of K, then $p^{\prime }$ has at least n−1 roots in $K_{\epsilon }$. This establishes, up to a logarithm, a conjecture of the first author: we also discuss an open problem whose solution would imply the full conjecture.

Keywords:

• Gauss-Lucas theorem
• roots
• zeroes
• derivatives
• critical points

AMS SUBJECT CLASSIFICATIONS:

• 26C10
• 30C15

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The authors have privately received a proof of a generalization of this conjecture from Vilmos Totik, who showed that the same conclusion holds for general measures (not just sums of point masses). As noted above, this in turn implies the conjectured approximate Gauss–Lucas theorem which originated in [16]. We look forward to seeing Totik's general result in print in the future.

Additional information

Funding

S.S. is supported by the Division of Mathematical Sciences (NSF) (DMS-1763179) and the Alfred P. Sloan Foundation.