Principal Submatrices, Restricted Invertibility, and a Quantitative Gauss–Lucas Theorem

Abstract

We apply the techniques developed by Marcus, Spielman, and Srivastava, working with principal submatrices in place of rank-$1$ decompositions to give an alternate proof of their results on restricted invertibility. This approach recovers results of theirs’ concerning the existence of well-conditioned column submatrices all the way up to the so-called modified stable rank. All constructions are algorithmic. The main novelty of this approach is that it leads to a new quantitative version of the classical Gauss–Lucas theorem on the critical points of complex polynomials. We show that for any degree $n$ polynomial $p$ and any $c \geq 1/2$, the area of the convex hull of the roots of $p^{(\lfloor cn \rfloor )}$ is at most $4(c-c^2)$ that of the area of the convex hull of the roots of $p$.

A Semicircle Law for Derivatives of Random Polynomials

“…There are a large number of results pertaining to the roots of p n and related properties (see [1,2,4,9,14,20,22,23,25,26,28,29,30,32,35,37,45,42,46,47,48,49,50]) as well as to the roots of higher derivatives (see [5,8,36]). Additionally, the analogous 'infinite degree' setting, in which polynomials are replaced by analytic functions, has also been considered -see Polya [34], Farmer & Rhoades [13] and Pemantle & Subramanian [33].…”

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

“…) shrinks to a single point. In 2018, M. Ravichandran [19] quantified the rate at which this nested sequence shrinks with the following theorem.…”

A note on restricted invertibility with weighted columns

“…Estimation in terms of stable rank. The estimation of the restricted invertibility principle is often stated in terms of stable rank in the literature [3,16,[19][20][21][25][26][27]. Here, we also present an estimation in terms of stable rank.…”

“…The algorithm in [16] runs in O(kmn θ+1 ) time. By directly applying this method to hermitian matrices and their principal matrices, Ravichandran [20] proved that for any k ≤ srank 4 (A), there is a subset S of size k such that…”