On the dynamics of the roots of polynomials under differentiation

Alazard, Thomas; Lazar, Omar; Nguyen, Quoc‐Hung

doi:10.1016/j.matpur.2022.04.001

Cited by 8 publications

(8 citation statements)

References 27 publications

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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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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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“…It would be nice to express F c in terms of classical zeta functions, but it is also useful to approximate it, near z = 0, by 1 z times a Taylor expansion…”

Section: Bi-periodic Functionsmentioning

confidence: 99%

“…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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“…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 [1,14,20,23] 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: All Roots Are Real Casementioning

confidence: 99%

“…It would be nice to express 𝐹 𝑐 in terms of classical zeta functions, but it is also useful to approximate it, near 𝑧 = 0, by 1 𝑧 times a Taylor expansion…”

Section: Bi-periodic Functionsmentioning

confidence: 99%

Modeling Complex Root Motion of Real Random Polynomials under Differentiation

Galligo

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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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 Δ𝑡 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. CCS CONCEPTS• Mathematics of computing → Partial differential equations.

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On the dynamics of the roots of polynomials under differentiation

Cited by 8 publications

References 27 publications

Global Solutions to the Tangential Peskin Problem in 2-D

Global Solutions to the Tangential Peskin Problem in 2-D

Modeling complex root motion of real random polynomials under differentiation

Modeling Complex Root Motion of Real Random Polynomials under Differentiation

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