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.
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Notes
See [57] for a modern formulation of Hilbert’s irreducibility theorem.
We thank Melanie Matchett Wood for describing the formulation and proof of this result.
Both of the applications mentioned here typically involve studying matrices other than the adjacency matrix of the underlying graph. For simplicity, we will only consider the adjacency matrix in this paper.
References
Adamczak, R., Chafaï, D., Wolff, P.: Circular law for random matrices with exchangeable entries. Random Struct. Algorithms 48(3), 454–479 (2016)
Aral, S., Walker, D.: Identifying influential and susceptible members of social networks. Science 337(6092), 337–341 (2012)
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)
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)
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)
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
Bourgain, J., Vu, V.H., Wood, P.M.: On the singularity probability of discrete random matrices. J. Funct. Anal. 258(2), 559–603 (2010)
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)
Chela, R.: Reducible polynomials. J. Lond. Math. Soc. 38, 183–188 (1963)
Cohen, S.D.: The distribution of the Galois groups of integral polynomials. Ill. J. Math. 23(1), 135–152 (1979)
Cohen, S.D.: The distribution of Galois groups and Hilbert’s irreducibility theorem. Proc. Lond. Math. Soc. 43(3), 227–250 (1981)
Cook, N.: On the singularity of adjacency matrices for random regular digraphs. arXiv:1411.0243, 9 Nov (2015)
Cook, N.: The circular law for signed random regular digraphs. arXiv:1508.00208, 2 Aug (2015)
Dietmann, R.: Probabilistic Galois theory. Bull. Lond. Math. Soc. 45(3), 453–462 (2013)
Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(4), 391–401 (1979)
Dummit, D.S., Foot, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken (2004)
Erdős, P., Rényi, A.: Asymmetric graphs. Acta Math. Hung. 14(3), 295–315 (1963)
Feldheim, O.N., Sen, A.: Double roots of random polynomials with integer coefficients. arXiv:1603.03811, 11 Mar (2016)
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)
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)
Godsil, C.: Controllable subsets in graphs. Ann. Comb. 16(4), 733–744 (2012)
Gu, S., Pasqualetti, F., Cieslak, M., Telesford, Q.K., et al.: Controllability of structural brain networks. Nat. Commun. 6, 8414 (2015)
Hespanha, J.P.: Linear Systems Theory. Princeton University Press, Princeton (2009)
Kalman, R.E.: Contributions to the theory of optimal control. Boletin de la Sociedad Matematica Mexicana 5, 102–119 (1960)
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)
Kalman, R.E.: Lectures on controllability and observability, pp. 1–151. C.I.M.E. Summer Schools, Cremonese, Rome (1969)
Kalman, R.E., Ho, Y.C., Narendra, K.S.: Controllability of linear dynamical systems. Contrib. Differ. Equ. 1(2), 189–213 (1962)
Konyagin, S.V.: On the number of irreducible polynomials with \(0,1\) coefficients. Acta Arith. 88(4), 333–350 (1999)
Knobloch, H.-W.: Zum Hilbertschen Irreduzibilitätssatz. Abh. Math. Sem. Univ. Hamburg 19, 176–190 (1955)
Knobloch, H.-W.: Die Seltenheit der reduziblen Polynome, Jber. Deutsch. Math. Verein. 59, Abt. 1, 12–19 (1956)
Kuba, G.: On the distribution of reducible polynomials. Math. Slovaca 59(3), 349–356 (2009)
Moree, P.: Artin’s primitive root conjecture—a survey. Integers 12(6), 1305–1416 (2012)
Naumov, A.A.: The elliptic law for random matrices, Vestnik Moskov. Univ. Ser. XV Vychisl. Mat. Kibernet. 2013, no. 1, 31–38, 48
Nguyen, H.H.: On the least singular value of random symmetric matrices. Electron. J. Probab. 17(53), 1–19 (2012)
Nguyen, H.H., O’Rourke, S.: The elliptic law, Int. Math. Res. Not. IMRN 2015, no. 17, 7620–7689
Nguyen, H.H., Vu, V.H.: Circular law for random discrete matrices of given row sum. J. Comb. 4(1), 1–30 (2013)
Odlyzko, A.M., Poonen, B.: Zeros of polynomials with \(0,1\) coefficients. L’Enseignement Mathématique 39, 317–348 (1993)
O’Rourke, S., Renfrew, D., Soshnikov, A., Vu, V.: Products of independent elliptic random matrices. J. Stat. Phys. 160(1), 89–119 (2015)
O’Rourke, S., Touri, B.: Controllability of random systems: universality and minimal controllability. arXiv:1506.03125, 9 Jun (2015)
O’Rourke, S., Touri, B.: On a conjecture of Godsil concerning controllable random graphs. arXiv:1511.05080, 16 Nov (2015)
Peled, R., Sen, A., Zeitouni, O.: Double roots of random Littlewood polynomials. Israel J. Math. 213(1), 55–77 (2016)
Pólya, G.: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen. Acta Math. 68, 145–254 (1937)
Pomerance, C.: Popular values of Euler’s function. Mathematika 27(1), 84–89 (1980)
Rivin, I.: Galois Groups of Generic Polynomials. arXiv:1511.06446, 19 Nov (2015)
Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)
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 Delhi
Tao, T., Vu, V.: Additive Combinatorics, Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)
Tao, T., Vu, V.: Local Universality of Zeroes of Random Polynomials. Int. Math. Res. Not. (2014). https://doi.org/10.1093/imrn/rnu084
Tao, T., Vu, V.: Random matrices: the circular law. Commun. Contemp. Math. 10(2), 261–307 (2008)
Tao, T., Vu, V.: Random matrices have simple spectrum. arXiv:1412.1438, 3 Dec (2014)
Terlov, G.: Low-degree factors of random polynomials with large integer coefficients. Work in progress
Vershynin, R.: Invertibility of symmetric random matrices. Random Struct. Algorithms 44, 135–182 (2014)
Vershynin, R.: Introduction to the non-asymptotic analysis of random matrices. In: Compressed Sensing, pp. 210–268. Cambridge University Press, Cambridge (2012)
van der Waerden, B.L.: Die Seltenheit der Gleichungen mit Affekt. Math. Ann. 109(1), 13–16 (1934)
van der Waerden, B.L.: Die Seltenheit der reduziblen Gleichungen und der Gleichungen mit Affekt. Monatsh. Math. Phys. 43(1), 133–147 (1936)
Weiss, B.L.: Probabilistic Galois theory over \(p\)-adic fields. J. Number Theory 133(5), 1537–1563 (2013)
Zywina, D.: Hilbert’s irreducibility theorem and the larger sieve. arXiv:1011.6465, 30 Nov (2010)
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.
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Philip Matchett Wood was partially supported by National Security Agency (NSA) Young Investigator Grant Numbers H98230-14-1-0149 and H98230-16-1-0301.
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O’Rourke, S., Wood, P.M. Low-Degree Factors of Random Polynomials. J Theor Probab 32, 1076–1104 (2019). https://doi.org/10.1007/s10959-018-0839-8
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DOI: https://doi.org/10.1007/s10959-018-0839-8