×

On the multiplicity of the zeros of polynomials with constrained coefficients. (English) Zbl 1507.30003

Rassias, Themistocles M. (ed.), Approximation theory and analytic inequalities. Cham: Springer. 165-177 (2021).
Author’s abstract: We survey a few recent results focusing on the multiplicity of the zero at 1 of polynomials with constrained coefficients. Some closely related problems and results are also discussed.
For the entire collection see [Zbl 1471.41001].

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
11C08 Polynomials in number theory
26C10 Real polynomials: location of zeros
Full Text: DOI

References:

[1] L.K. Hua, Introduction to Number Theory(Springer, Berlin/Heidelberg/New York, 1982) · Zbl 0483.10001
[2] K.G. Hare, J. Jankauskas, On Newman and Littlewood polynomials with a prescribed number of zeros inside the unit disk, Math. Comput. (published electronically: October 27, 2020) · Zbl 1461.11141
[3] G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers(Clarendon Press, Oxford, 1938) · JFM 64.0093.03
[4] C.S. Güntürk, Approximation by power series with ± 1 coefficients. Int. Math. Res. Not. 26, 1601-1610 (2005) · Zbl 1079.60015 · doi:10.1155/IMRN.2005.1601
[5] W. Foster, I. Krasikov, An improvement of a Borwein-Erdélyi-Kós result. Methods Appl. Anal. 7(4), 605-614 (2000) · Zbl 1009.41005
[6] L.B.O. Ferguson, Approximation by Polynomials with Integral Coefficients(American Mathematical Society, Providence, 1980) · Zbl 0441.41003 · doi:10.1090/surv/017
[7] P. Erdős, P. Turán, On the distribution of roots of polynomials. Ann. Math. 57, 105-119 (1950) · Zbl 0036.01501
[8] T. Erdélyi, Pseudo-Boolean functions and the multiplicity of the zeros of polynomials. J. Anal. Math. 127(1), 91-108 (2015) · Zbl 1331.30005 · doi:10.1007/s11854-015-0025-1
[9] T. Erdélyi, Problem 11437, Zeros of polynomials with unit coefficients. Am. Math. Mon. 116(5), 464 (2009)
[10] T. Erdélyi, An improvement of the Erdős-Turán theorem on the zero distribution of polynomials. C. R. Acad. Sci. Paris Sér. I Math. 346, 267-270 (2008) · Zbl 1156.30006 · doi:10.1016/j.crma.2008.01.020
[11] T. Erdélyi, Extensions of the Bloch-Pólya theorem on the number of distinct real zeros of polynomials, Journal de théorie des nombres de Bordeaux 20, 281-287 (2008) · Zbl 1163.11022 · doi:10.5802/jtnb.627
[12] T. Erdélyi, Polynomials with Littlewood-type coefficient constraints, in Approximation Theory X: Abstract and Classical Analysis, ed. by C.K. Chui, L.L. Schumaker, J. Stöckler (Vanderbilt University Press, Nashville, 2002), pp. 153-196 · Zbl 1053.41008
[13] T. Erdélyi, Markov-Bernstein type inequalities for polynomials under Erdős-type constraints, in Paul Erdős and his Mathematics I, ed. by G. Halász, L. Lovász, D. Miklós, V.T. Sós. Bolyai Society Mathematical Studies, vol. 11 (Springer, New York, 2002), pp. 219-239 · Zbl 1032.41007
[14] M. Dudik, L.J. Schulman, Reconstruction from subsequences. J. Comb. Theory Ser. A 103(2), 337-348 (2003) · Zbl 1039.68098 · doi:10.1016/S0097-3165(03)00103-1
[15] A. Dubickas, J. Jankauskas, On Newman polynomials which divide no Littlewood polynomial. Math. Comput. 78(265), 327-344 (2009) · Zbl 1208.11123 · doi:10.1090/S0025-5718-08-02138-8
[16] A. Dubickas, Polynomials with multiple roots at 1. Int. J. Number Theory 10(2), 391-400 (2014) · Zbl 1307.11033 · doi:10.1142/S1793042113501005
[17] A. Dubickas, Three problems of polynomials of small measure. Acta Arith. 98, 279-292 (2001) · Zbl 0972.11102 · doi:10.4064/aa98-3-5
[18] A. Dubickas, On the order of vanishing at 1 of a polynomial. Lithuanian Math. J. 39, 365-370 (1999) · Zbl 0963.12001 · doi:10.1007/BF02465586
[19] P. Drungilas, J. Jankauskas, J. Šiurys, On Littlewood and Newman polynomial multiples of Borwein polynomials. Math. Comput. 87(311), 1523-1541 (2018) · Zbl 1390.11122 · doi:10.1090/mcom/3258
[20] P. Drungilas, J. Jankauskas, G. Junevičius, L. Klebonas, J. Šiurys, On certain multiples of Littlewood and Newman polynomials. Bull. Korean Math. Soc. 55(5), 1491-1501 (2018) · Zbl 1444.11044
[21] E. Croot, D. Hart, h-fold sums from a set with few products. SIAM J. Discret. Math. 24(2), 505-519 (2010) · Zbl 1221.11202
[22] D. Coppersmith, T.J. Rivlin, The growth of polynomials bounded at equally spaced points. SIAM J. Math. Anal. 23(4), 970-983 (1992) · Zbl 0769.26003 · doi:10.1137/0523054
[23] J.M. Cooper, A.M. Dutle, Greedy Galois games. Am. Math. Mon. 120(5), 441451 (2013) · Zbl 1272.91015
[24] P.G. Casazza, N.J. Kalton, Roots of complex polynomials and Weyl-Heisenberg frame sets. Proc. Am. Math. Soc. 130(8), 2313-2318 (2002) · Zbl 0991.42023 · doi:10.1090/S0002-9939-02-06352-9
[25] H. Buhrman, R. Cleve, R. deWolf, C. Zalka, Bounds for small-error and zero-error quantum algorithms, in 40th Annual Symposium on Foundations of Computer Science, New York (IEEE Computer Society, Los Alamitos, 1999), pp. 358-368
[26] D.W. Boyd, On a problem of Byrnes concerning polynomials with restricted coefficients. Math. Comput. 66, 1697-1703 (1997) · Zbl 0893.12004 · doi:10.1090/S0025-5718-97-00892-2
[27] P. Borwein, M.J. Mossinghoff, Polynomials with height 1 and prescribed vanishing at 1. Exp. Math. 9(3), 425-433 (2000) · Zbl 0999.12001 · doi:10.1080/10586458.2000.10504419
[28] P. Borwein, C. Ingalls, The Prouhet, Tarry, Escott problem. Ens. Math. 40, 3-27 (1994) · Zbl 0810.11016
[29] P. Borwein, T. Erdélyi, J. Zhang, Müntz systems and orthogonal Müntz-Legendre polynomials. Trans. Am. Math. Soc. 342, 523-542 (1992) · Zbl 0799.41015
[30] P. Borwein, T. Erdélyi, F. Littmann, Polynomials with coefficients from a finite set. Trans. Am. Math. Soc. 360, 5145-5154 (2008) · Zbl 1158.30002 · doi:10.1090/S0002-9947-08-04605-9
[31] P. Borwein, T. Erdélyi, G. Kós, The multiplicity of the zero at 1 of polynomials with constrained coefficients. Acta Arithm. 159(4), 387-395 (2013) · Zbl 1284.11054 · doi:10.4064/aa159-4-7
[32] P. Borwein, T. Erdélyi, G. Kós, Littlewood-type problems on [0, 1]. Proc. Lond. Math. Soc. 79, 22-46 (1999) · Zbl 1039.11046 · doi:10.1112/S0024611599011831
[33] P. Borwein, T. Erdélyi, R. Ferguson, R. Lockhart, On the zeros of cosine polynomial: an old problem of Littlewood. Ann. Math. (2) 167, 1109-1117 (2008) · Zbl 1186.11045
[34] P. Borwein, T. Erdélyi, Lower bounds for the number of zeros of cosine polynomials in the period: a problem of Littlewood. Acta Arith. 128, 377-384 (2007) · Zbl 1124.41009 · doi:10.4064/aa128-4-5
[35] P. Borwein, T. Erdélyi, Trigonometric polynomials with many real zeros and a littlewood-type problem. Proc. Am. Math. Soc. 129(3), 725-730 (2001) · Zbl 0968.41008 · doi:10.1090/S0002-9939-00-06021-4
[36] P.B. Borwein, T. Erdélyi, Littlewood-type problems on subarcs of the unit circle. Indiana Univ. Math. J. 46, 1323-1346 (1997) · Zbl 0930.30005 · doi:10.1512/iumj.1997.46.1435
[37] P.B. Borwein T. Erdélyi, Generalizations of Müntz’s theorem via a Remez-type inequality for Müntz spaces. J. Am. Math. Soc. 10, 327-329 (1997) · Zbl 0864.41014 · doi:10.1090/S0894-0347-97-00225-7
[38] P. Borwein, T. Erdélyi, On the zeros of polynomials with restricted coefficients. Illinois J. Math. 41, 667-675 (1997) · Zbl 0906.30005 · doi:10.1215/ijm/1256068987
[39] P. Borwein, T. Erdélyi, Questions about polynomials with 0,-1,+1 coefficients. Constr. Approx. 12(3), 439-442 (1996) · Zbl 0877.41013
[40] P. Borwein, T. Erdélyi, The integer Chebyshev problem. Math. Comput. 65, 661-681 (1996) · Zbl 0859.11044 · doi:10.1090/S0025-5718-96-00702-8
[41] P. Borwein, T. Erdélyi, Polynomials and Polynomial Inequalities(Springer, New York, 1995) · Zbl 0840.26002 · doi:10.1007/978-1-4612-0793-1
[42] P. Borwein, Computational Excursions in Analysis and Number Theory(Springer, New York, 2002) · Zbl 1020.12001 · doi:10.1007/978-0-387-21652-2
[43] E. Bombieri, J. Vaaler, Polynomials with low height and prescribed vanishing, in Analytic Number Theory and Diophantine Problems(Birkhäuser, Boston, MA, 1987), pp. 53-73 · Zbl 0629.10024
[44] A. Bloch, G. Pólya, On the roots of certain algebraic equations. Proc. Lond. Math. Soc 33, 102-114 (1932) · JFM 57.0128.03 · doi:10.1112/plms/s2-33.1.102
[45] F. Beaucoup, P. Borwein, D.W. Boyd, C. Pinner, Multiple roots of [-1, 1] power series. J. Lond. Math. Soc. (2) 57, 135-147 (1998) · Zbl 0922.30004
[46] B. Aparicio, New bounds on the minimal Diophantine deviation from zero on [0, 1] and [0, 1/4]. Actus Sextas J. Mat. Hisp.-Lusitanas 289-291 (1979) · Zbl 0938.11501
[47] F. Amoroso, Sur le diamètre transfini entier d’un intervalle réel. Ann. Inst. Fourier, Grenoble 40, 885-911 (1990) · Zbl 0713.41004
[48] P · Zbl 0544.10045
[49] V. Totik, P. Varjú, Polynomials with prescribed zeros and small norm. Acta Sci. Math. (Szeged) 73(3-4), 593-611 (2007) · Zbl 1174.26013
[50] G. Szegő, Bemerkungen zu einem Satz von E. Schmidt uber algebraische Gleichungen. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 86-98 (1934) · JFM 60.0061.01
[51] K. Soundararajan, Equidistribution of zeros of polynomials. Am. Math. Mon. 126(3), 226236 (2019) · Zbl 1409.42003
[52] I.E. Shparlinski, Finite Fields: Theory and Computation - The Meeting Point of Number Theory, Computer Science, Coding Theory and Cryptography, Dordrecht/London, 1999 · Zbl 0967.11052
[53] I. Schur, Untersuchungen über algebraische Gleichungen. Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. 403-428 (1933) · JFM 59.0911.02
[54] F. Rodier, Sur la non-linéarité des fonctions booléennes. Acta Arith. 115(1), 1-22 (2004) · Zbl 1115.94011 · doi:10.4064/aa115-1-1
[55] E.A. Rakhmanov, Bounds for polynomials with a unit discrete norm. Ann. Math. 165, 55-88 (2007) · Zbl 1124.41014 · doi:10.4007/annals.2007.165.55
[56] I.E. Pritsker, A.A. Sola, Expected discrepancy for zeros of random algebraic polynomials. Proc. Am. Math. Soc. 142, 4251-4263 (2014) · Zbl 1301.30006 · doi:10.1090/S0002-9939-2014-12147-2
[57] C. Pinner, Double roots of [−1, 1] power series and related matters. Math. Comput. 68(2), 1149-1178 (1999) · Zbl 0920.30008 · doi:10.1090/S0025-5718-99-01042-X
[58] M. Pathiaux, Sur les multiples de polynômes irréductibles associés à certains nombres algébriques, 9 pp., Séminaire Delange-Pisot-Poitou 14 (1972-1973) · Zbl 0323.12001
[59] A.M. Odlyzko, B. Poonen, Zeros of polynomials with 0, 1 coefficients. Enseign. Math. 39, 317-348 (1993) · Zbl 0814.30006
[60] N. Nisan, M. Szegedy, On the degree of Boolean functions as real polynomials. Earlier version in STOC92. Comput. Complex. 4(4), 301-313 (1994) · Zbl 0829.68047
[61] M.J. Mossinghoff, Polynomials with restricted coefficients and prescribed noncyclotomic factors, (electronic). Lond. Math. Soc. J. Comput. Math. 6, 314-325 (2003) · Zbl 1134.11319
[62] M. Minsky, S. Papert, Perceptrons: An Introduction to Computational Geometry(MIT Press, Cambridge, MA, 1968) · Zbl 0197.43702
[63] M. Mignotte, Sur les multiples des polynmes irrductibles. Bull. Soc. Math. Belg. 27, 225-229 (1975) · Zbl 0405.12021
[64] I. Krasikov, Multiplicity of zeros and discrete orthogonal polynomials. Results Math. 45(1-2), 59-66 (2004) · Zbl 1052.30012 · doi:10.1007/BF03322997
[65] G. Kós, P. Ligeti, P. Sziklai, Reconstruction of matrices from submatrices. Math. Comput. 78, 1733-1747 (2009) · Zbl 1198.05023 · doi:10.1090/S0025-5718-09-02210-8
[66] V.V. Andrievskii, H-P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation(Springer, New York, 2002) · Zbl 0995.30001 · doi:10.1007/978-1-4757-4999-1
[67] T. Erdélyi, Coppersmith-Rivlin type inequalities and the order of vanishing of polynomials at 1. Acta Arith. 172(3), 271-284 (2016) · Zbl 1345.26021
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.