×

A nonlocal transport equation describing roots of polynomials under differentiation. (English) Zbl 1431.35196

The author considers the distribution of the zeroes of a polynomial of degree \(n\) with \(n\) real zeroes under iterated differentiation. In a mean-field approximation he derives formally a nonlinear transport equation for the description of the evolution of the distribution under iterated differentiation. The arcsine distribution, the family of semicircle distributions, and a family of solutions contained in the Marchenko-Pastur law are shown to solve the nonlinear transport equation.

MSC:

35Q70 PDEs in connection with mechanics of particles and systems of particles
44A15 Special integral transforms (Legendre, Hilbert, etc.)
26C10 Real polynomials: location of zeros
31A99 Two-dimensional potential theory
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A50 Best approximation, Chebyshev systems
33C52 Orthogonal polynomials and functions associated with root systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Balodis, Pedro; C\'{o}rdoba, Antonio, An inequality for Riesz transforms implying blow-up for some nonlinear and nonlocal transport equations, Adv. Math., 214, 1, 1-39 (2007) · Zbl 1133.35078 · doi:10.1016/j.aim.2006.07.021
[2] Blower, Gordon, Random matrices: high dimensional phenomena, London Mathematical Society Lecture Note Series 367, x+437 pp. (2009), Cambridge University Press, Cambridge · Zbl 1278.60003 · doi:10.1017/CBO9781139107129
[3] Bochner, Salomon, Book Review: Gesammelte Schriften, Bull. Amer. Math. Soc. (N.S.), 1, 6, 1020-1022 (1979) · doi:10.1090/S0273-0979-1979-14724-4
[4] Bosbach, Christof; Gawronski, Wolfgang, Strong asymptotics for Laguerre polynomials with varying weights, J. Comput. Appl. Math.. Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997), 99, 1-2, 77-89 (1998) · Zbl 0933.33010 · doi:10.1016/S0377-0427(98)00147-2
[5] de Bruijn, N. G., On the zeros of a polynomial and of its derivative, Nederl. Akad. Wetensch., Proc., 49, 1037-1044 = Indagationes Math. 8, 635-642 (1946) (1946) · Zbl 0060.05604
[6] de Bruijn, N. G.; Springer, T. A., On the zeros of a polynomial and of its derivative. II, Nederl. Akad. Wetensch., Proc., 50, 264-270=Indagationes Math. 9, 458-464 (1947) (1947) · Zbl 0029.19801
[7] Carrillo, Jos\'{e} A.; Ferreira, Lucas C. F.; Precioso, Juliana C., A mass-transportation approach to a one dimensional fluid mechanics model with nonlocal velocity, Adv. Math., 231, 1, 306-327 (2012) · Zbl 1252.35224 · doi:10.1016/j.aim.2012.03.036
[8] Castro, A.; C\'{o}rdoba, D., Global existence, singularities and ill-posedness for a nonlocal flux, Adv. Math., 219, 6, 1916-1936 (2008) · Zbl 1186.35002 · doi:10.1016/j.aim.2008.07.015
[9] Chae, Dongho; C\'{o}rdoba, Antonio; C\'{o}rdoba, Diego; Fontelos, Marco A., Finite time singularities in a 1D model of the quasi-geostrophic equation, Adv. Math., 194, 1, 203-223 (2005) · Zbl 1128.76372 · doi:10.1016/j.aim.2004.06.004
[10] coif R. Coifman and S. Steinerberger, A Remark on the Arcsine Distribution and the Hilbert transform, arXiv:1810.10128. · Zbl 1446.44002
[11] Constantin, P.; Lax, P. D.; Majda, A., A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38, 6, 715-724 (1985) · Zbl 0615.76029 · doi:10.1002/cpa.3160380605
[12] C\'{o}rdoba, Antonio; C\'{o}rdoba, Diego; Fontelos, Marco A., Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2), 162, 3, 1377-1389 (2005) · Zbl 1101.35052 · doi:10.4007/annals.2005.162.1377
[13] \'{C}urgus, Branko; Mascioni, Vania, A contraction of the Lucas polygon, Proc. Amer. Math. Soc., 132, 10, 2973-2981 (2004) · Zbl 1050.30006 · doi:10.1090/S0002-9939-04-07231-4
[14] Dette, H.; Studden, W. J., Some new asymptotic properties for the zeros of Jacobi, Laguerre, and Hermite polynomials, Constr. Approx., 11, 2, 227-238 (1995) · Zbl 0898.41003 · doi:10.1007/BF01203416
[15] Dimitrov, Dimitar K., A refinement of the Gauss-Lucas theorem, Proc. Amer. Math. Soc., 126, 7, 2065-2070 (1998) · Zbl 0895.30006 · doi:10.1090/S0002-9939-98-04381-0
[16] Do, Tam; Hoang, Vu; Radosz, Maria; Xu, Xiaoqian, One-dimensional model equations for hyperbolic fluid flow, Nonlinear Anal., 140, 1-11 (2016) · Zbl 1381.35135 · doi:10.1016/j.na.2016.03.002
[17] Dong, Hongjie, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal., 255, 11, 3070-3097 (2008) · Zbl 1170.35004 · doi:10.1016/j.jfa.2008.08.005
[18] Dong, Hongjie; Li, Dong, On a one-dimensional \(\alpha \)-patch model with nonlocal drift and fractional dissipation, Trans. Amer. Math. Soc., 366, 4, 2041-2061 (2014) · Zbl 1302.35334 · doi:10.1090/S0002-9947-2013-06075-8
[19] Elstrodt, J\"{u}rgen, Partialbruchentwicklung des Kotangens, Herglotz-Trick und die Weierstra\ss sche stetige, nirgends differenzierbare Funktion, Math. Semesterber., 45, 2, 207-220 (1998) · Zbl 0949.39010 · doi:10.1007/s005910050046
[20] Erd\H{o}s, Paul; Tur\'{a}n, Paul, On interpolation. III. Interpolatory theory of polynomials, Ann. of Math. (2), 41, 510-553 (1940) · Zbl 0024.39102 · doi:10.2307/1968733
[21] Erd\H{o}s, P.; Freud, G., On orthogonal polynomials with regularly distributed zeros, Proc. London Math. Soc. (3), 29, 521-537 (1974) · Zbl 0294.33006 · doi:10.1112/plms/s3-29.3.521
[22] Farmer, David W.; Rhoades, Robert C., Differentiation evens out zero spacings, Trans. Amer. Math. Soc., 357, 9, 3789-3811 (2005) · Zbl 1069.30005 · doi:10.1090/S0002-9947-05-03721-9
[23] gauss C.F.Gauss, Werke, Band 3, G\"ottingen 1866, S. 120:112.
[24] Gawronski, Wolfgang, Strong asymptotics and the asymptotic zero distributions of Laguerre polynomials \(L_n^{(an+\alpha)}\) and Hermite polynomials \(H_n^{(an+\alpha)}\), Analysis, 13, 1-2, 29-67 (1993) · Zbl 0810.33002 · doi:10.1524/anly.1993.13.12.29
[25] granero R. Granero-Belinchon, On a nonlocal differential equation describing roots of polynomials under differentiation, arXiv:1812.00082. · Zbl 1401.35315
[26] Hanin, Boris, Pairing of zeros and critical points for random polynomials, Ann. Inst. Henri Poincar\'{e} Probab. Stat., 53, 3, 1498-1511 (2017) · Zbl 1373.30006 · doi:10.1214/16-AIHP767
[27] Kabluchko, Zakhar, Critical points of random polynomials with independent identically distributed roots, Proc. Amer. Math. Soc., 143, 2, 695-702 (2015) · Zbl 1314.30008 · doi:10.1090/S0002-9939-2014-12258-1
[28] Kornyik, Mikl\'{o}s; Michaletzky, Gy\"{o}rgy, On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law, Ann. Univ. Sci. Budapest. Sect. Comput., 46, 137-151 (2017) · Zbl 1399.15047
[29] Lazar, Omar; Lemari\'{e}-Rieusset, Pierre-Gilles, Infinite energy solutions for a 1D transport equation with nonlocal velocity, Dyn. Partial Differ. Equ., 13, 2, 107-131 (2016) · Zbl 1350.35062 · doi:10.4310/DPDE.2016.v13.n2.a2
[30] Li, Dong; Rodrigo, Jos\'{e} L., On a one-dimensional nonlocal flux with fractional dissipation, SIAM J. Math. Anal., 43, 1, 507-526 (2011) · Zbl 1231.35172 · doi:10.1137/100794924
[31] lucas F. Lucas, Sur une application de la M\'ecanique rationnelle \`“a la th\'”eorie des \'equations, in: Comptes Rendus de l’Acad\'emie des Sciences (89), Paris 1979, S. 224-226. · JFM 11.0068.01
[32] Malamud, S. M., Inverse spectral problem for normal matrices and the Gauss-Lucas theorem, Trans. Amer. Math. Soc., 357, 10, 4043-4064 (2005) · Zbl 1080.15012 · doi:10.1090/S0002-9947-04-03649-9
[33] Mart\'{\i}nez-Finkelshtein, Andrei; Mart\'{\i}nez-Gonz\'{a}lez, Pedro; Orive, Ram\'{o}n, On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters, J. Comput. Appl. Math.. Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 133, 1-2, 477-487 (2001) · Zbl 0990.33009 · doi:10.1016/S0377-0427(00)00654-3
[34] O’Rourke, Sean; Williams, Noah, Pairing between zeros and critical points of random polynomials with independent roots, Trans. Amer. Math. Soc., 371, 4, 2343-2381 (2019) · Zbl 1426.30006 · doi:10.1090/tran/7496
[35] Pemantle, Robin; Rivin, Igor, The distribution of zeros of the derivative of a random polynomial. Advances in combinatorics, 259-273 (2013), Springer, Heidelberg · Zbl 1272.30004
[36] rav M. Ravichandran, Principal submatrices, restricted invertibility and a quantitative Gauss-Lucas theorem, arXiv:1609.04187. · Zbl 1487.15030
[37] Silvestre, Luis; Vicol, Vlad, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc., 368, 9, 6159-6188 (2016) · Zbl 1334.35254 · doi:10.1090/tran6651
[38] Steinerberger, Stefan, Electrostatic interpretation of zeros of orthogonal polynomials, Proc. Amer. Math. Soc., 146, 12, 5323-5331 (2018) · Zbl 1464.34049 · doi:10.1090/proc/14226
[39] steini2 S. Steinerberger, A Stability Version of the Gauss-Lucas Theorem and Applications, to appear in J. Austral. Math. Soc. · Zbl 1446.26012
[40] riesz A. Stoyanoff, Sur un Theorem de M. Marcel Riesz, Nouv. Annal. de Mathematique 1 (1926), 97-99. · JFM 52.0098.03
[41] Totik, Vilmos, The Gauss-Lucas theorem in an asymptotic sense, Bull. Lond. Math. Soc., 48, 5, 848-854 (2016) · Zbl 1359.26018 · doi:10.1112/blms/bdw047
[42] Ullman, J. L., On the regular behaviour of orthogonal polynomials, Proc. London Math. Soc. (3), 24, 119-148 (1972) · Zbl 0232.33007 · doi:10.1112/plms/s3-24.1.119
[43] Van Assche, Walter, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics 1265, vi+201 pp. (1987), Springer-Verlag, Berlin · Zbl 0617.42014 · doi:10.1007/BFb0081880
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.