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Uniform irreducibility and Galois groups of generalized Laguerre polynomials

A mix of human-written and AI-generated textHuman understanding: some partsmath.GM — General Mathematicsmath.NT — Number Theory

Contributed by Khai-Hoan Nguyen-Dang ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We prove that the generalized Laguerre polynomial Hn,sH_{n,s} is irreducible over Q\mathbb Q and has Galois group containing AnA_n for every positive integer ss, once nn exceeds an effective absolute constant. Equivalently, every numerator and denominator in the exponential Pad\'e table has these properties once its own degree is sufficiently large. The exceptional positive integral pairs form an effectively computable finite set. The proof combines integral traces at degenerate fibers, Pad\'e resultants, and an energy inequality for Galois-invariant positive semidefinite forms on the roots.

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References

  1. P. Banerjee and R. Bera, Classifying families of orthogonal polynomials having Galois group the alternating group, J. Number Theory 240 (2022), 31--49. doi:10.1016/j.jnt.2021.12.009 (https://doi.org/10.1016/j.jnt.2021.12.009).DOI
  2. P. Banerjee, M. Filaseta, C. E. Finch and J. R. Leidy, On classifying Laguerre polynomials which have Galois group the alternating group, J. Théor. Nombres Bordeaux 25 (2013), no. 1, 1--30. doi:10.5802/jtnb.822 (https://doi.org/10.5802/jtnb.822).DOI
  3. L. Bary-Soroker and O. Ben-Porath, On the Galois theory of generalized Laguerre polynomials and trimmed exponential, Acta Arith. 200 (2021), no. 2, 183--196. doi:10.4064/aa200825-7-3 (https://doi.org/10.4064/aa200825-7-3).DOI
  4. L. Bary-Soroker and B. Kadets, Generic Manin--Mumford , preprint (2026), arXiv:2609.09354 (https://arxiv.org/abs/2609.09354).arXiv
  5. J. Borcea and P. Brändén, Pólya--Schur master theorems for circular domains and their boundaries, Ann. of Math. (2) 170 (2009), 465--492. doi:10.4007/annals.2009.170.465 (https://doi.org/10.4007/annals.2009.170.465).DOI
  6. M. G. de Bruin, E. B. Saff, and R. S. Varga, On the zeros of generalized Bessel polynomials. I, Indag. Math. (Proceedings) 84 (1981), no. 1, 1--13. doi:10.1016/1385-7258(81)90013-5 (https://doi.org/10.1016/1385-7258(81)90013-5).DOI
  7. T. C. Burness and R. M. Guralnick, Fixed point ratios for finite primitive groups and applications, Adv. Math. 411 (2022), 108778. doi:10.1016/j.aim.2022.108778 (https://doi.org/10.1016/j.aim.2022.108778).DOI
  8. G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863--911. doi:10.1080/00927878308822884 (https://doi.org/10.1080/00927878308822884).DOI
  9. K. Castillo, Complete Bernstein functions and scaled ultraspherical zeros, preprint (2026), arXiv:2609.19186 (https://arxiv.org/abs/2609.19186).arXiv
  10. R. F. Coleman, On the Galois groups of the exponential Taylor polynomials, Enseign. Math. (2) 33 (1987), no. 3--4, 183--189. doi:10.5169/seals-87891 (https://doi.org/10.5169/seals-87891).DOI
Show all 48 referencesShow fewer references
  1. J. Cullinan and N. Scheel, On the arithmetic of Padé approximants to the exponential function, J. Ramanujan Math. Soc. 37 (2022), no. 3, 207--219. https://www.mathjournals.org/jrms/2022-037-003/2022-037-003-001.html.link
  2. Z. Deng, A proof of the irreducibility conjecture for Legendre polynomials, preprint (2026), arXiv:2609.22336 (https://arxiv.org/abs/2609.22336).arXiv
  3. J. D. Dixon and B. Mortimer, Permutation Groups , Graduate Texts in Mathematics, vol. 163, Springer, New York, 1996. doi:10.1007/978-1-4612-0731-3 (https://doi.org/10.1007/978-1-4612-0731-3).DOI
  4. P. Dusart, Explicit estimates of some functions over primes, Ramanujan J. 45 (2018), no. 1, 227--251. doi:10.1007/s11139-016-9839-4 (https://doi.org/10.1007/s11139-016-9839-4).DOI
  5. P. Dusart, Estimates of some functions over primes without R.H., preprint (2010), arXiv:1002.0442 (https://arxiv.org/abs/1002.0442).arXiv
  6. P. Erdős, On consecutive integers, Nieuw Arch. Wisk. (3) 3 (1955), 124--128.
  7. P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois J. Math. 19 (1975), no. 2, 292--301. https://users.renyi.hu/~p_erdos/1975-46.pdf.link
  8. P. Erdős and J. Turk, Products of integers in short intervals, Acta Arith. 44 (1984), no. 2, 147--174. doi:10.4064/aa-44-2-147-174 (https://doi.org/10.4064/aa-44-2-147-174).DOI
  9. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349--366. doi:10.1007/BF01388432 (https://doi.org/10.1007/BF01388432).DOI
  10. K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. I, Proc. Natl. Acad. Sci. USA 35 (1949), no. 11, 652--655. doi:10.1073/pnas.35.11.652 (https://doi.org/10.1073/pnas.35.11.652).DOI
  11. M. Filaseta, The irreducibility of all but finitely many Bessel polynomials, Acta Math. 174 (1995), no. 2, 383--397. doi:10.1007/BF02392470 (https://doi.org/10.1007/BF02392470).DOI
  12. M. Filaseta and T.-Y. Lam, On the irreducibility of the generalized Laguerre polynomials, Acta Arith. 105 (2002), no. 2, 177--182. doi:10.4064/aa105-2-4 (https://doi.org/10.4064/aa105-2-4).DOI
  13. M. Filaseta, S. Laishram, and N. Saradha, Solving n(n+d)⋯(n+(k−1)d)=by2n(n+d)\cdots(n+(k-1)d)=by^2 with P(b)≤CkP(b)\le Ck, Int. J. Number Theory 8 (2012), 161--173. doi:10.1142/S1793042112500091 (https://doi.org/10.1142/S1793042112500091).DOI
  14. M. Filaseta and O. Trifonov, The irreducibility of the Bessel polynomials, J. Reine Angew. Math. 550 (2002), 125--140. doi:10.1515/crll.2002.069 (https://doi.org/10.1515/crll.2002.069).DOI
  15. K. Ford, The distribution of integers with a divisor in a given interval, Ann. of Math. (2) 168 (2008), no. 2, 367--433. doi:10.4007/annals.2008.168.367 (https://doi.org/10.4007/annals.2008.168.367).DOI
  16. E. Grosswald, On some algebraic properties of the Bessel polynomials, Trans. Amer. Math. Soc. 71 (1951), no. 2, 197--210. doi:10.1090/S0002-9947-1951-0053280-2 (https://doi.org/10.1090/S0002-9947-1951-0053280-2).DOI
  17. E. Grosswald, Bessel Polynomials , Lecture Notes in Mathematics, vol. 698, Springer, Berlin--New York, 1978. doi:10.1007/BFb0063135 (https://doi.org/10.1007/BFb0063135).DOI
  18. F. Hajir, On the Galois group of generalized Laguerre polynomials, J. Théor. Nombres Bordeaux 17 (2005), 517--525. doi:10.5802/jtnb.505 (https://doi.org/10.5802/jtnb.505).DOI
  19. F. Hajir, Algebraic properties of a family of generalized Laguerre polynomials, Canad. J. Math. 61 (2009), no. 3, 583--603. doi:10.4153/CJM-2009-031-6 (https://doi.org/10.4153/CJM-2009-031-6).DOI
  20. F. Hajir and S. Wong, Specializations of one-parameter families of polynomials, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1127--1163. doi:10.5802/aif.2208 (https://doi.org/10.5802/aif.2208).DOI
  21. A. Jindal and S. Laishram, Families of Laguerre polynomials with alternating group as Galois group, J. Number Theory 241 (2022), 387--429. doi:10.1016/j.jnt.2022.04.001 (https://doi.org/10.1016/j.jnt.2022.04.001).DOI
  22. A. Jindal, S. Laishram, and R. Sarma, Irreducibility and Galois groups of generalized Laguerre polynomials Ln(−1−n−r)(x)L_n^{(-1-n-r)}(x), J. Number Theory 183 (2018), 388--406. doi:10.1016/j.jnt.2017.08.003 (https://doi.org/10.1016/j.jnt.2017.08.003).DOI
  23. A. Jindal, S. G. Nair, and T. N. Shorey, Extension of irreducibility results on generalized Laguerre polynomials Ln(−n−s−1)(x)L_n^{(-n-s-1)}(x), Publ. Math. Debrecen 105 (2024), no. 3--4, 473--494. doi:10.5486/PMD.2024.9906 (https://doi.org/10.5486/PMD.2024.9906).DOI
  24. A. Levin, Variations on a theme of Runge: effective determination of integral points on certain varieties, J. Théor. Nombres Bordeaux 20 (2008), no. 2, 385--417. doi:10.5802/jtnb.634 (https://doi.org/10.5802/jtnb.634).DOI
  25. D. Lombardo, On the ℓ\ell-adic Galois representations attached to nonsimple abelian varieties, Ann. Inst. Fourier (Grenoble) 66 (2016), no. 3, 1217--1245. doi:10.5802/aif.3035 (https://doi.org/10.5802/aif.3035).DOI
  26. J. H. Loxton, Some problems involving powers of integers, Acta Arith. 46 (1986), no. 2, 113--123. doi:10.4064/aa-46-2-113-123 (https://doi.org/10.4064/aa-46-2-113-123).DOI
  27. A. W. Marcus, D. A. Spielman, and N. Srivastava, Finite free convolutions of polynomials, Probab. Theory Related Fields 182 (2022), 807--848. doi:10.1007/s00440-021-01105-w (https://doi.org/10.1007/s00440-021-01105-w).DOI
  28. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II, Izv. Math. 64 (2000), no. 6, 1217--1269. doi:10.1070/IM2000v064n06ABEH000314 (https://doi.org/10.1070/IM2000v064n06ABEH000314).DOI
  29. J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28 (1952), no. 4, 177--181. doi:10.3792/pja/1195570997 (https://doi.org/10.3792/pja/1195570997).DOI
  30. NIST Digital Library of Mathematical Functions, Chapter 18, Orthogonal Polynomials, especially §§18.5, 18.7--18.8, 18.9, and 18.34. https://dlmf.nist.gov/18 (accessed 4 November 2026).link
  31. P. Rabarison, F. Pazuki, and P. Molin, Exponentielle tronquée et autres contes galoisiens, Publ. Math. Besançon Algèbre Théor. Nombres (2024), 105--117. doi:10.5802/pmb.57 (https://doi.org/10.5802/pmb.57).DOI
  32. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), no. 1, 64--94. doi:10.1215/ijm/1255631807 (https://doi.org/10.1215/ijm/1255631807).DOI
  33. I. Schur, Einige Sätze über Primzahlen mit Anwendungen auf Irreduzibilitätsfragen. I, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl. (1929), 125--136.
  34. I. Schur, Gleichungen ohne Affekt, Sitzungsber. Preuss. Akad. Wiss., Phys.-Math. Kl. (1930), 443--449.
  35. I. Schur, Affektlose Gleichungen in der Theorie der Laguerreschen und Hermiteschen Polynome, J. Reine Angew. Math. 165 (1931), 52--58. doi:10.1515/crll.1931.165.52 (https://doi.org/10.1515/crll.1931.165.52).DOI
  36. C. L. Stewart, On heights of multiplicatively dependent algebraic numbers, Acta Arith. 133 (2008), no. 2, 97--108. doi:10.4064/aa133-2-1 (https://doi.org/10.4064/aa133-2-1).DOI
  37. Y. G. Zarhin, Very simple 22-adic representations and hyperelliptic jacobians, Mosc. Math. J. 2 (2002), no. 2, 403--431. doi:10.17323/1609-4514-2002-2-2-403-431 (https://doi.org/10.17323/1609-4514-2002-2-2-403-431).DOI
  38. Y. G. Zarhin, Homomorphisms of hyperelliptic Jacobians, Proc. Steklov Inst. Math. 241 (2003), 79--92. https://www.mathnet.ru/eng/tm389.link

Version history

  1. v1Submitted by Khai-Hoan Nguyen-DangInitial depositCurrentNov 04, 2026