A nonlocal transport equation describing roots of polynomials under differentiation

Steinerberger, Stefan

doi:10.1090/proc/14699

A nonlocal transport equation describing roots of polynomials under differentiation
HTML articles powered by AMS MathViewer

by Stefan Steinerberger

Proc. Amer. Math. Soc. 147 (2019), 4733-4744

DOI: https://doi.org/10.1090/proc/14699

Published electronically: July 30, 2019

PDF | Request permission

Abstract:

Let $p_n$ be a polynomial of degree $n$ having all its roots on the real line distributed according to a smooth function $u(0,x)$. One could wonder how the distribution of roots behaves under iterated differentation of the function, i.e., how the density of roots of $p_n^{(k)}$ evolves. We derive a nonlinear transport equation with nonlocal flux \begin{equation*} u_t + \frac {1}{\pi }\left ( \arctan { \left ( \frac {Hu}{ u}\right )} \right )_x = 0 \qquad \text {on} ~\operatorname {supp} \left \{u>0\right \}, \end{equation*} where $H$ is the Hilbert transform. This equation has three very different compactly supported solutions: (1) the arcsine distribution $u(t,x) = (1-x^2)^{-1/2} \chi _{(-1,1)}$, (2) the family of semicircle distributions \begin{equation*} u(t,x) = \frac {2}{\pi } \sqrt {(T-t) - x^2}, \end{equation*} and (3) a family of solutions contained in the Marchenko–Pastur law.

References

Pedro Balodis and Antonio Córdoba, An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations, Adv. Math. 214 (2007), no. 1, 1–39. MR 2348021, DOI 10.1016/j.aim.2006.07.021
Gordon Blower, Random matrices: high dimensional phenomena, London Mathematical Society Lecture Note Series, vol. 367, Cambridge University Press, Cambridge, 2009. MR 2566878, DOI 10.1017/CBO9781139107129
Salomon Bochner, Book Review: Gesammelte Schriften, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 1020–1022. MR 1567203, DOI 10.1090/S0273-0979-1979-14724-4
Christof Bosbach and Wolfgang Gawronski, Strong asymptotics for Laguerre polynomials with varying weights, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 1998, pp. 77–89. MR 1662685, DOI 10.1016/S0377-0427(98)00147-2
N. G. de Bruijn, On the zeros of a polynomial and of its derivative, Nederl. Akad. Wetensch., Proc. 49 (1946), 1037–1044 = Indagationes Math. 8, 635–642 (1946). MR 19157
N. G. de Bruijn and T. A. Springer, On the zeros of a polynomial and of its derivative. II, Nederl. Akad. Wetensch., Proc. 50 (1947), 264–270=Indagationes Math. 9, 458–464 (1947). MR 21148
José A. Carrillo, Lucas C. F. Ferreira, and Juliana C. Precioso, A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math. 231 (2012), no. 1, 306–327. MR 2935390, DOI 10.1016/j.aim.2012.03.036
A. Castro and D. Córdoba, Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math. 219 (2008), no. 6, 1916–1936. MR 2456270, DOI 10.1016/j.aim.2008.07.015
Dongho Chae, Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math. 194 (2005), no. 1, 203–223. MR 2141858, DOI 10.1016/j.aim.2004.06.004
R. Coifman and S. Steinerberger, A Remark on the Arcsine Distribution and the Hilbert transform, arXiv:1810.10128.
P. Constantin, P. D. Lax, and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 (1985), no. 6, 715–724. MR 812343, DOI 10.1002/cpa.3160380605
Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377–1389. MR 2179734, DOI 10.4007/annals.2005.162.1377
Branko Ćurgus and Vania Mascioni, A contraction of the Lucas polygon, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2973–2981. MR 2063118, DOI 10.1090/S0002-9939-04-07231-4
H. Dette and W. J. Studden, Some new asymptotic properties for the zeros of Jacobi, Laguerre, and Hermite polynomials, Constr. Approx. 11 (1995), no. 2, 227–238. MR 1342385, DOI 10.1007/BF01203416
Dimitar K. Dimitrov, A refinement of the Gauss-Lucas theorem, Proc. Amer. Math. Soc. 126 (1998), no. 7, 2065–2070. MR 1452801, DOI 10.1090/S0002-9939-98-04381-0
Tam Do, Vu Hoang, Maria Radosz, and Xiaoqian Xu, One-dimensional model equations for hyperbolic fluid flow, Nonlinear Anal. 140 (2016), 1–11. MR 3492724, DOI 10.1016/j.na.2016.03.002
Hongjie Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal. 255 (2008), no. 11, 3070–3097. MR 2464570, DOI 10.1016/j.jfa.2008.08.005
Hongjie Dong and Dong Li, On a one-dimensional $\alpha$-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2041–2061. MR 3152722, DOI 10.1090/S0002-9947-2013-06075-8
Jürgen Elstrodt, Partialbruchentwicklung des Kotangens, Herglotz-Trick und die Weierstraßsche stetige, nirgends differenzierbare Funktion, Math. Semesterber. 45 (1998), no. 2, 207–220 (German, with German summary). MR 1684563, DOI 10.1007/s005910050046
Paul Erdös and Paul Turán, On interpolation. III. Interpolatory theory of polynomials, Ann. of Math. (2) 41 (1940), 510–553. MR 1999, DOI 10.2307/1968733
P. Erdős and G. Freud, On orthogonal polynomials with regularly distributed zeros, Proc. London Math. Soc. (3) 29 (1974), 521–537. MR 420119, DOI 10.1112/plms/s3-29.3.521
David W. Farmer and Robert C. Rhoades, Differentiation evens out zero spacings, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3789–3811. MR 2146650, DOI 10.1090/S0002-9947-05-03721-9
C. F. Gauss, Werke, Band 3, Göttingen 1866, S. 120:112.
Wolfgang Gawronski, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials $L_n^{(an+\alpha )}$ and Hermite polynomials $H_n^{(an+\alpha )}$, Analysis 13 (1993), no. 1-2, 29–67. MR 1245742, DOI 10.1524/anly.1993.13.12.29
R. Granero-Belinchon, On a nonlocal differential equation describing roots of polynomials under differentiation, arXiv:1812.00082.
Boris Hanin, Pairing of zeros and critical points for random polynomials, Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017), no. 3, 1498–1511 (English, with English and French summaries). MR 3689975, DOI 10.1214/16-AIHP767
Zakhar Kabluchko, Critical points of random polynomials with independent identically distributed roots, Proc. Amer. Math. Soc. 143 (2015), no. 2, 695–702. MR 3283656, DOI 10.1090/S0002-9939-2014-12258-1
Miklós Kornyik and György Michaletzky, On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law, Ann. Univ. Sci. Budapest. Sect. Comput. 46 (2017), 137–151. MR 3722670
Omar Lazar and Pierre-Gilles Lemarié-Rieusset, Infinite energy solutions for a 1D transport equation with nonlocal velocity, Dyn. Partial Differ. Equ. 13 (2016), no. 2, 107–131. MR 3520809, DOI 10.4310/DPDE.2016.v13.n2.a2
Dong Li and José L. Rodrigo, On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal. 43 (2011), no. 1, 507–526. MR 2783214, DOI 10.1137/100794924
F. Lucas, Sur une application de la Mécanique rationnelle à la théorie des équations, in: Comptes Rendus de l’Académie des Sciences (89), Paris 1979, S. 224–226.
S. M. Malamud, Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4043–4064. MR 2159699, DOI 10.1090/S0002-9947-04-03649-9
Andrei Martínez-Finkelshtein, Pedro Martínez-González, and Ramón Orive, On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 477–487. MR 1858305, DOI 10.1016/S0377-0427(00)00654-3
Sean O’Rourke and Noah Williams, Pairing between zeros and critical points of random polynomials with independent roots, Trans. Amer. Math. Soc. 371 (2019), no. 4, 2343–2381. MR 3896083, DOI 10.1090/tran/7496
Robin Pemantle and Igor Rivin, The distribution of zeros of the derivative of a random polynomial, Advances in combinatorics, Springer, Heidelberg, 2013, pp. 259–273. MR 3363974
M. Ravichandran, Principal submatrices, restricted invertibility and a quantitative Gauss-Lucas theorem, arXiv:1609.04187.
Luis Silvestre and Vlad Vicol, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6159–6188. MR 3461030, DOI 10.1090/tran6651
Stefan Steinerberger, Electrostatic interpretation of zeros of orthogonal polynomials, Proc. Amer. Math. Soc. 146 (2018), no. 12, 5323–5331. MR 3866871, DOI 10.1090/proc/14226
S. Steinerberger, A Stability Version of the Gauss-Lucas Theorem and Applications, to appear in J. Austral. Math. Soc.
A. Stoyanoff, Sur un Theorem de M. Marcel Riesz, Nouv. Annal. de Mathematique 1 (1926), 97–99.
Vilmos Totik, The Gauss-Lucas theorem in an asymptotic sense, Bull. Lond. Math. Soc. 48 (2016), no. 5, 848–854. MR 3556367, DOI 10.1112/blms/bdw047
J. L. Ullman, On the regular behaviour of orthogonal polynomials, Proc. London Math. Soc. (3) 24 (1972), 119–148. MR 291718, DOI 10.1112/plms/s3-24.1.119
Walter Van Assche, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol. 1265, Springer-Verlag, Berlin, 1987. MR 903848, DOI 10.1007/BFb0081880