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Journal of Theoretical Probability

, Volume 32, Issue 2, pp 1076–1104 | Cite as

Low-Degree Factors of Random Polynomials

  • Sean O’Rourke
  • Philip Matchett WoodEmail author
Article
  • 48 Downloads

Abstract

We study the probability that a monic polynomial with integer coefficients has a low-degree factor over the integers, which is equivalent to having a low-degree algebraic root. It is known in certain cases that random polynomials with integer coefficients are very likely to be irreducible, and our project can be viewed as part of a general program of testing whether this is a universal behavior exhibited by many random polynomial models. Our main result shows that pointwise delocalization of the roots of a random polynomial can be used to imply that the polynomial is unlikely to have a low-degree factor over the integers. We apply our main result to a number of models of random polynomials, including characteristic polynomials of random matrices, where strong delocalization results are known.

Keywords

Random polynomials Irreducible Random matrices Delocalization 

Mathematics Subject Classification (2010)

11C08 15B52 

Notes

Acknowledgements

We thank Melanie Matchett Wood for many useful conversations and for contributing key ideas for Theorem 1.7. The first author thanks Peter D.T.A. Elliott, Richard Green, and Katherine Stange for useful discussions and references. The second author thanks Van Vu for originally suggesting this line of inquiry. The authors also thank the anonymous referee for useful comments and suggestions which led to the current version of Theorem 2.1. Christian Borst, Evan Boyd, Claire Brekken, and Samantha Solberg produced Fig. 1 and were supported by NSF Grant DMS-1301690 and co-supervised by Melanie Matchett Wood. The second author thanks Steve Goldstein for helping direct Borst, Boyd, Brekken, and Solberg’s research. The second author also thanks the Simons Foundation for providing Magma licenses and the Center for High Throughput Computing (CHTC) at the University of Wisconsin-Madison for providing computer resources.

References

  1. 1.
    Adamczak, R., Chafaï, D., Wolff, P.: Circular law for random matrices with exchangeable entries. Random Struct. Algorithms 48(3), 454–479 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aral, S., Walker, D.: Identifying influential and susceptible members of social networks. Science 337(6092), 337–341 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babai, L.: Automorphism groups, isomorphism, reconstruction. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Chapter 27 of the Handbook of Combinatorics, vol. 2, pp. 1447–1540. North Holland Elsevier, Amsterdam (1995)Google Scholar
  4. 4.
    Bary-Soroker, L., Kozma, G.: Is a bivariate polynomial with \(\pm 1\) coefficients irreducible? Very likely!. Int. J. Number Theory 13(4), 933–936 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bond, R.M., Fariss, C.J., Jones, J.J., Kramer, A.D.I., et al.: A 61-million-person experiment in social influence and political mobilization. Nature 489(7415), 295–298 (2012)CrossRefGoogle Scholar
  6. 6.
    Borst, C., Boyd, E., Brekken, C., Solberg, S., Wood, M.M., Wood, P.M.: Irreducibility of random polynomials. arXiv:1705.03709, 10 May 2017. To appear in Experimental Mathematics
  7. 7.
    Bourgain, J., Vu, V.H., Wood, P.M.: On the singularity probability of discrete random matrices. J. Funct. Anal. 258(2), 559–603 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chan, A., Godsil, C.D.: Symmetry and eigenvectors. Chapter 3 of Graph Symmetry: Algebraic Methods and Applications, Volume 497 of the series NATO ASI Series pp. 75–106 (edited by G. Hahn and G. Sabidussi) (1997)Google Scholar
  9. 9.
    Chela, R.: Reducible polynomials. J. Lond. Math. Soc. 38, 183–188 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohen, S.D.: The distribution of the Galois groups of integral polynomials. Ill. J. Math. 23(1), 135–152 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cohen, S.D.: The distribution of Galois groups and Hilbert’s irreducibility theorem. Proc. Lond. Math. Soc. 43(3), 227–250 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cook, N.: On the singularity of adjacency matrices for random regular digraphs. arXiv:1411.0243, 9 Nov (2015)
  13. 13.
    Cook, N.: The circular law for signed random regular digraphs. arXiv:1508.00208, 2 Aug (2015)
  14. 14.
    Dietmann, R.: Probabilistic Galois theory. Bull. Lond. Math. Soc. 45(3), 453–462 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(4), 391–401 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dummit, D.S., Foot, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken (2004)Google Scholar
  17. 17.
    Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Hung. 14(3), 295–315 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Feldheim, O.N., Sen, A.: Double roots of random polynomials with integer coefficients. arXiv:1603.03811, 11 Mar (2016)
  19. 19.
    Fox, M.D., Halko, M.A., Eldaief, M.C., Pascual-Leone, A.: Measuring and manipulating brain connectivity with resting state functional connectivity magnetic resonance imaging (fcMRI) and transcranial magnetic stimulation (TMS). Neuroimage 62(4), 2232–2243 (2012)CrossRefGoogle Scholar
  20. 20.
    Gallagher, P.X.: The large sieve and probabilistic Galois theory, Analytic number theory. In: Proceedings of Symposium Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 91–101. Am. Math. Soc., Providence, R.I (1973)Google Scholar
  21. 21.
    Godsil, C.: Controllable subsets in graphs. Ann. Comb. 16(4), 733–744 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gu, S., Pasqualetti, F., Cieslak, M., Telesford, Q.K., et al.: Controllability of structural brain networks. Nat. Commun. 6, 8414 (2015)CrossRefGoogle Scholar
  23. 23.
    Hespanha, J.P.: Linear Systems Theory. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  24. 24.
    Kalman, R.E.: Contributions to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana 5, 102–119 (1960)MathSciNetGoogle Scholar
  25. 25.
    Kalman, R.E.: On the general theory of control systems. In: Proceedings of 1st IFAC Congress, Moscow 1960, Vol. 1, pp. 481–492. Butterworth, London (1961)Google Scholar
  26. 26.
    Kalman, R.E.: Lectures on controllability and observability, pp. 1–151. C.I.M.E. Summer Schools, Cremonese, Rome (1969)Google Scholar
  27. 27.
    Kalman, R.E., Ho, Y.C., Narendra, K.S.: Controllability of linear dynamical systems. Contrib. Differ. Equ. 1(2), 189–213 (1962)MathSciNetGoogle Scholar
  28. 28.
    Konyagin, S.V.: On the number of irreducible polynomials with \(0,1\) coefficients. Acta Arith. 88(4), 333–350 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Knobloch, H.-W.: Zum Hilbertschen Irreduzibilitätssatz. Abh. Math. Sem. Univ. Hamburg 19, 176–190 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Knobloch, H.-W.: Die Seltenheit der reduziblen Polynome, Jber. Deutsch. Math. Verein. 59, Abt. 1, 12–19 (1956)Google Scholar
  31. 31.
    Kuba, G.: On the distribution of reducible polynomials. Math. Slovaca 59(3), 349–356 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Moree, P.: Artin’s primitive root conjecture—a survey. Integers 12(6), 1305–1416 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Naumov, A.A.: The elliptic law for random matrices, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 2013, no. 1, 31–38, 48Google Scholar
  34. 34.
    Nguyen, H.H.: On the least singular value of random symmetric matrices. Electron. J. Probab. 17(53), 1–19 (2012)MathSciNetGoogle Scholar
  35. 35.
    Nguyen, H.H., O’Rourke, S.: The elliptic law, Int. Math. Res. Not. IMRN 2015, no. 17, 7620–7689Google Scholar
  36. 36.
    Nguyen, H.H., Vu, V.H.: Circular law for random discrete matrices of given row sum. J. Comb. 4(1), 1–30 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Odlyzko, A.M., Poonen, B.: Zeros of polynomials with \(0,1\) coefficients. L’Enseignement Mathématique 39, 317–348 (1993)MathSciNetzbMATHGoogle Scholar
  38. 38.
    O’Rourke, S., Renfrew, D., Soshnikov, A., Vu, V.: Products of independent elliptic random matrices. J. Stat. Phys. 160(1), 89–119 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    O’Rourke, S., Touri, B.: Controllability of random systems: universality and minimal controllability. arXiv:1506.03125, 9 Jun (2015)
  40. 40.
    O’Rourke, S., Touri, B.: On a conjecture of Godsil concerning controllable random graphs. arXiv:1511.05080, 16 Nov (2015)
  41. 41.
    Peled, R., Sen, A., Zeitouni, O.: Double roots of random Littlewood polynomials. Israel J. Math. 213(1), 55–77 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Pólya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen. Acta Math. 68, 145–254 (1937)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Pomerance, C.: Popular values of Euler’s function. Mathematika 27(1), 84–89 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rivin, I.: Galois Groups of Generic Polynomials. arXiv:1511.06446, 19 Nov (2015)
  45. 45.
    Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Rudelson, M., Vershynin, R.: Non-asymptotic theory of random matrices: extreme singular values. In: Proceedings of the International Congress of Mathematicians. Volume III, pp. 1576–1602, Hindustan Book Agency, New DelhiGoogle Scholar
  47. 47.
    Tao, T., Vu, V.: Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  48. 48.
    Tao, T., Vu, V.: Local Universality of Zeroes of Random Polynomials. Int. Math. Res. Not. (2014).  https://doi.org/10.1093/imrn/rnu084 zbMATHGoogle Scholar
  49. 49.
    Tao, T., Vu, V.: Random matrices: the circular law. Commun. Contemp. Math. 10(2), 261–307 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Tao, T., Vu, V.: Random matrices have simple spectrum. arXiv:1412.1438, 3 Dec (2014)
  51. 51.
    Terlov, G.: Low-degree factors of random polynomials with large integer coefficients. Work in progressGoogle Scholar
  52. 52.
    Vershynin, R.: Invertibility of symmetric random matrices. Random Struct. Algorithms 44, 135–182 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Compressed Sensing, pp. 210–268. Cambridge University Press, Cambridge (2012)Google Scholar
  54. 54.
    van der Waerden, B.L.: Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109(1), 13–16 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    van der Waerden, B.L.: Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt. Monatsh. Math. Phys. 43(1), 133–147 (1936)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Weiss, B.L.: Probabilistic Galois theory over \(p\)-adic fields. J. Number Theory 133(5), 1537–1563 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Zywina, D.: Hilbert’s irreducibility theorem and the larger sieve. arXiv:1011.6465, 30 Nov (2010)

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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