The Flow of Polynomial Roots Under Differentiation

Kiselev, A.; Tan, C.Y.

doi:10.48550/arxiv.2012.09080

Cited by 5 publications

(8 citation statements)

References 65 publications

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“…This text has been mainly written, but for various reasons not completely finished already in Spring 2018; its content has been presented during a workshop "Hausdorff geometry of polynomials and polynomial sequences" at the Mittag-Leffler institute in Stockholm. Since then several relevant papers discussing similar questions about the behavior of roots of polynomials under consecutive differentiations appeared, see e.g., [St1,St2,HoKa,KiTa]. In particular, paper [St2] contains a heuristic deduction of an intriguing partial differential equation satisfied (under several additional assumptions) by the density of roots under differentiation.…”

Section: Resultsmentioning

confidence: 99%

Rodrigues' descendants of a polynomial and Boutroux curves

Bøgvad¹,

Hägg²,

Shapiro³

2021

Preprint

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Motivated by the classical Rodrigues' formula, we study below the root asymptotic of the polynomial sequencedz [αn] , n = 0, 1, . . . where P (z) is a fixed univariate polynomial, α is a fixed positive number smaller than deg P , and [αn] stands for the integer part of αn.Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in [Be, BM]. As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by {R [αn],n,P (z)} as well as higher derivatives of powers of more general functions.platt och avintetgjord släpar jag nollan min vid håret in i oändlighet.

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Section: Resultsmentioning

confidence: 99%

Rodrigues' descendants of a polynomial and Boutroux curves

Bøgvad¹,

Hägg²,

Shapiro³

2021

Preprint

View full text Add to dashboard Cite

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“…Also see [31,32]. Recently, its well-posedness was proved in [33,34,35]. Although (1.13) looks similar to (1.1) if one ignores the denominator of the right-hand side, its solutions can behave differently from those to (1.1) because of very different form of nonlinearity and degeneracy.…”

Section: Definition 12 (Dissipative Weak Solution) Under the Addition...mentioning

confidence: 99%

Global Solutions to the Tangential Peskin Problem in 2-D

Tong¹

2022

Preprint

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We introduce the tangential Peskin problem in 2-D, which is a scalar drift-diffusion equation with a nonlocal drift. It is derived with a new Eulerian perspective from a special setting of the 2-D Peskin problem where an infinitely long and straight 1-D elastic string deforms tangentially in the Stokes flow induced by itself in the plane. For initial datum in the energy class satisfying natural weak assumptions, we prove existence of its global solutions. This is considered as a supercritical problem in the existing analysis of the Peskin problem based on Lagrangian formulations. Regularity and long-time behavior of the constructed solution is established. Uniqueness of the solution is proved under additional assumptions.

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“…He provided a rather informal but inspiring construction of his PDE: following the classical electrical interpretation of a critical point of P n as an equilibrium of repulsion-attraction forces from the roots of P n , he divided them into a local near field (with a local uniformity property assumption) and an averaged far field estimated via a Cauchy-Stieltjes integral, hence a Hilbert transform of the density. Then, several articles [14,23,20,1] successively offered more and more complicated and detailed analysis in the periodic setting (i.e. on the circle), they provided rigorous proof of "crystallization" under repeated differentiation.…”

Section: Introductionmentioning

confidence: 99%

Modeling complex root motion of real random polynomials under differentiation

Galligo

2022

Preprint

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In this paper, we consider nonlocal, nonlinear partial differential equations to model anisotropic dynamics of complex root sets of random polynomials under differentiation. These equations aim to generalise the recent PDE obtained by Stefan Steinerberger (2019) in the real case, and the PDE obtained by Sean O'Rourke and Stefan Steinerberger (2020) in the radial case, which amounts to work in 1D. These PDEs approximate dynamics of the complex roots for random polynomials of sufficiently high degree n. The unit of the time t corresponds to n differentiations, and the increment ∆t corresponds to 1 n . The general situation in 2D, in particular for complex roots of real polynomials, was not yet addressed. The purpose of this paper is to present a first attempt in that direction. We assume that the roots are distributed according to a regular distribution with a local homogeneity property (defined in the text), and that this property is maintained under differentiation. This allows us to derive a system of two coupled equations to model the motion. Our system could be interesting for other applications. The paper is illustrated with examples computed with the Maple system.

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The Flow of Polynomial Roots Under Differentiation

Cited by 5 publications

References 65 publications

Rodrigues' descendants of a polynomial and Boutroux curves

Rodrigues' descendants of a polynomial and Boutroux curves

Global Solutions to the Tangential Peskin Problem in 2-D

Modeling complex root motion of real random polynomials under differentiation

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