Displaying items by tag: Polynomial roots

Alexander Kiselev, and Changhui Tan

SIAM Journal on Mathematical Analysis, Volume 54, No. 3, pp. 3161-3191 (2022).

Abstract

In this paper, we analyze a nonlocal nonlinear partial differential equation formally derived by Stefan Steinerberger to model dynamics of roots of polynomials under differentiation. This partial differential equation is critical and bears striking resemblance to hydrodynamic models used to described collective behavior of agents (such as birds, fish or robots) in mathematical biology. We consider periodic setting and show global regularity and exponential in time convergence to uniform density for solutions corresponding to strictly positive smooth initial data.

	doi:10.1137/21M1422859
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264

Alexander Kiselev, and Changhui Tan

Annals of PDE, Volume 8, No. 2, Article 16, 69pp. (2022)

Abstract

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian \((-\Delta)^{1/2}\).

	doi:10.1007/s40818-022-00135-4
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	This work is supported by NSF grant DMS #1853001 and DMS #2108264