HEXAGONbeta

Galois groups of Taylor polynomials: prescriptions and arithmetic families

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

Contributed by Khai-Hoan Nguyen-Dang ↗

Version 3 / Oct 26, 2026 / CC BY 4.0

Abstract

Let fn(x)=∑j=0najxjf_n(x)=\sum_{j=0}^n a_jx^j be the Taylor polynomials of a power series with nonzero rational coefficients. We study the proportions of degrees for which the Galois group over $\Q$ is symmetric, alternating, or neither. We prove that the set of degrees n≥3n\geq3 with symmetric group can be chosen arbitrarily, even for an entire function with positive rational coefficients whose Taylor polynomials have distinct negative real roots. More precisely, at each degree n≥3n\geq3 outside the chosen set, fnf_n has exactly one rational root, and the group on the other roots is the full symmetric group. Thus the proportions need not converge; when they do, the proportion of other groups can be any number in [0,1][0,1]. These conclusions answer negatively two questions of Katz and Rivin. Moreover, for fixed $s\in\Q\setminus\Z$, we also give a condition on the prime divisors of rational multipliers of the coefficients s/(s+j)s/(s+j) that forces symmetric groups in density one. For the Taylor polynomials of (1−x)−s(1-x)^{-s}, we extend Wei's eventual group classification to fixed number fields. Over $\Q$, the number of exceptions to the symmetric group up to degree XX is csX+Os(1)c_s\sqrt X+O_s(1), with an explicit constant cs≥0c_s\geq0. We also determine the groups of the central-binomial Taylor polynomials in every degree and prove that every nonconstant Catalan Taylor polynomial has symmetric group. This proves Sun's Catalan group conjecture and his irreducible-reduction criterion for the monic reciprocals of the central-binomial polynomials.

Provenance statement

OpenAI's ChatGPT was used in the preparation and revision of this paper to assist with mathematical arguments and to prepare and run exact-arithmetic code for the finite certificates. The October 2026 revisions used GPT-6 Astra Pro.

Tools used

OpenAI
ChatGPTVersion Astra

References

  1. M. Allen and M. Filaseta, A generalization of a second irreducibility theorem of I. Schur , Acta Arith. 109 (2003), 65--79. doi:10.4064/aa109-1-3 (https://doi.org/10.4064/aa109-1-3).DOI
  2. 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). arXiv:2008.05165 (https://arxiv.org/abs/2008.05165).arXivDOI
  3. L. Bary-Soroker, D. Koukoulopoulos, and G. Kozma, Irreducibility of random polynomials: general measures , Invent. Math. 233 (2023), 1041--1120. doi:10.1007/s00222-023-01193-6 (https://doi.org/10.1007/s00222-023-01193-6).DOI
  4. J. P. Bell, The upper density of an automatic set is rational , J. Théor. Nombres Bordeaux 32 (2020), 585--604. doi:10.5802/jtnb.1135 (https://doi.org/10.5802/jtnb.1135).DOI
  5. M. A. Bennett, G. Martin, K. O'Bryant, and A. Rechnitzer, Explicit bounds for primes in arithmetic progressions , Illinois J. Math. 62 (2018), 427--532. doi:10.1215/ijm/1552442669 (https://doi.org/10.1215/ijm/1552442669).DOI
  6. G. Blachar, E. Breuillard, and G. Kozma, Galois groups of random polynomials of large degree , preprint (2026), arXiv:2609.40123v1 (https://arxiv.org/abs/2609.40123v1).arXiv
  7. A. Borisov, M. Filaseta, T. Y. Lam, and O. Trifonov, Classes of polynomials having only one non-cyclotomic irreducible factor , Acta Arith. 90 (1999), 121--153. doi:10.4064/aa-90-2-121-153 (https://doi.org/10.4064/aa-90-2-121-153).DOI
  8. D. Brink, On alternating and symmetric groups as Galois groups , Israel J. Math. 142 (2004), 47--60. doi:10.1007/BF02771527 (https://doi.org/10.1007/BF02771527).DOI
  9. A. Chambert-Loir, The theorem of Jentzsch--Szegő on an analytic curve: application to the irreducibility of truncations of power series , Int. J. Number Theory 7 (2011), no. 7, 1807--1823. doi:10.1142/S1793042111004691 (https://doi.org/10.1142/S1793042111004691). arXiv:1004.3813 (https://arxiv.org/abs/1004.3813).arXivDOI
  10. H. Cohen, A Course in Computational Algebraic Number Theory , Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. doi:10.1007/978-3-662-02945-9 (https://doi.org/10.1007/978-3-662-02945-9).DOI
Show all 52 referencesShow fewer references
  1. R. F. Coleman, On the Galois groups of the exponential Taylor polynomials , Enseign. Math. (2) 33 (1987), 183--189. doi:10.5169/seals-87891 (https://doi.org/10.5169/seals-87891).DOI
  2. B. Conrad, Math 248A: Tame ramification and composite fields , course handout, Stanford University, Theorem 2.1. https://virtualmath1.stanford.edu/~conrad/248APage/handouts/tamecomp.pdf.link
  3. J. Cullinan and R. Gajek-Leonard, On the discriminants of truncated logarithmic polynomials , Integers 25 (2025), Paper A39. doi:10.5281/zenodo.15283716 (https://doi.org/10.5281/zenodo.15283716).DOI
  4. J. Cullinan, F. Hajir, and E. Sell, Algebraic properties of a family of Jacobi polynomials , J. Théor. Nombres Bordeaux 21 (2009), no. 1, 97--108. doi:10.5802/jtnb.659 (https://doi.org/10.5802/jtnb.659).DOI
  5. NIST Digital Library of Mathematical Functions , Chapters 7, 13, 16, and 18, accessed 23 October 2026. https://dlmf.nist.gov/.link
  6. P. Dusart, Explicit estimates of some functions over primes , Ramanujan J. 45 (2018), 227--251. doi:10.1007/s11139-016-9839-4 (https://doi.org/10.1007/s11139-016-9839-4).DOI
  7. P. Erdős, F. Luca, and C. Pomerance, On the proportion of numbers coprime to a given integer , in Anatomy of Integers , CRM Proceedings & Lecture Notes, vol. 46, American Mathematical Society, 2008, 47--64.
  8. G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , Invent. Math. 73 (1983), no. 3, 349--366. doi:10.1007/BF01388432 (https://doi.org/10.1007/BF01388432); erratum, 75 (1984), 381. doi:10.1007/BF01388572 (https://doi.org/10.1007/BF01388572).DOI
  9. M. Filaseta and R. Moy, On the Galois group over $\Q$ of a truncated binomial expansion , Colloq. Math. 154 (2018), 295--308. doi:10.4064/cm7474-3-2018 (https://doi.org/10.4064/cm7474-3-2018).DOI
  10. M. Filaseta and I. Solan, An extension of a theorem of Ljunggren , Math. Scand. 84 (1999), 5--10. doi:10.7146/math.scand.a-13928 (https://doi.org/10.7146/math.scand.a-13928).DOI
  11. H. Garmi, Galois Groups of Trinomials with Small Coefficients , preprint (2026), arXiv:2610.22764v2 (https://arxiv.org/abs/2610.22764v2).arXiv
  12. F. Hajir, On the Galois group of generalized Laguerre polynomials , J. Théor. Nombres Bordeaux 17 (2005), no. 2, 517--525. doi:10.5802/jtnb.505 (https://doi.org/10.5802/jtnb.505).DOI
  13. D. R. Heath-Brown, A mean value estimate for real character sums , Acta Arith. 72 (1995), no. 3, 235--275. doi:10.4064/aa-72-3-235-275 (https://doi.org/10.4064/aa-72-3-235-275).DOI
  14. J. I. Hutchinson, On a remarkable class of entire functions , Trans. Amer. Math. Soc. 25 (1923), no. 3, 325--332. doi:10.1090/S0002-9947-1923-1501248-1 (https://doi.org/10.1090/S0002-9947-1923-1501248-1).DOI
  15. C. G. Jockusch, Jr., and P. E. Schupp, Generic computability, Turing degrees, and asymptotic density , J. London Math. Soc. (2) 85 (2012), 472--490. doi:10.1112/jlms/jdr051 (https://doi.org/10.1112/jlms/jdr051).DOI
  16. G. A. Jones, Primitive permutation groups containing a cycle , Bull. Aust. Math. Soc. 89 (2014), 159--165. doi:10.1017/S000497271300049X (https://doi.org/10.1017/S000497271300049X).DOI
  17. N. M. Katz and I. Rivin, Speculations in Galois theory , to appear in Algebraic Geometry and the Langlands Program: A Volume in Honour of Gérard Laumon , Simons Symposia, Springer, Cham. https://web.math.princeton.edu/~nmk/kGalois_11bis.pdf.link
  18. B. Koley and A. S. Reddy, An irreducibility criterion for polynomials over integers , Bull. Math. Soc. Sci. Math. Roumanie 63(111) (2020), no. 1, 83--89. https://ssmr.ro/bulletin/pdf/63-1/articol_6.pdf.link
  19. V. P. Kostov and B. Shapiro, Hardy--Petrovitch--Hutchinson's problem and partial theta function , Duke Math. J. 162 (2013), no. 5, 825--861. doi:10.1215/00127094-2087264 (https://doi.org/10.1215/00127094-2087264). arXiv:1106.6262 (https://arxiv.org/abs/1106.6262).arXivDOI
  20. D. Koukoulopoulos, The Distribution of Prime Numbers , Graduate Studies in Mathematics, vol. 203, American Mathematical Society, Providence, RI, 2019. doi:10.1090/gsm/203 (https://doi.org/10.1090/gsm/203).DOI
  21. R. Levingston and D. E. Taylor, The theorem of Marggraff on primitive permutation groups which contain a cycle , Bull. Austral. Math. Soc. 15 (1976), 125--128. doi:10.1017/S0004972700036832 (https://doi.org/10.1017/S0004972700036832).DOI
  22. J. S. Milne, Algebraic Number Theory , version 3.08, 19 July 2020. https://www.jmilne.org/math/CourseNotes/ANT.pdf.link
  23. H. L. Montgomery and R. C. Vaughan, The large sieve , Mathematika 20 (1973), 119--134. doi:10.1112/S0025579300004708 (https://doi.org/10.1112/S0025579300004708).DOI
  24. P. Müller, Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12 (2013), 369--438. doi:10.2422/2036-2145.201012_002 (https://doi.org/10.2422/2036-2145.201012_002).DOI
  25. J. Nagura, On the interval containing at least one prime number , Proc. Japan Acad. 28 (1952), 177--181. doi:10.3792/pja/1195570997 (https://doi.org/10.3792/pja/1195570997).DOI
  26. K.-H. Nguyen-Dang, Arithmetic of Jacobi polynomials , preprint, 22 October 2026, hexagon:2610.00468v1 (https://hexagonmath.org/2610.00468v1).hexagon
  27. A. M. Odlyzko, Lower bounds for discriminants of number fields. II , Tohoku Math. J. (2) 29 (1977), no. 2, 209--216. doi:10.2748/tmj/1178240652 (https://doi.org/10.2748/tmj/1178240652).DOI
  28. M. Peter, The asymptotic distribution of elements in automatic sequences , Theoret. Comput. Sci. 301 (2003), 285--312. doi:10.1016/S0304-3975(02)00587-X (https://doi.org/10.1016/S0304-3975(02)00587-X).DOI
  29. P. Rabarison, F. Pazuki, and P. Molin, Exponentielle tronquée et autres contes galoisiens , Publ. Math. Besançon Algèbre Théorie Nombres (2024), 105--117. doi:10.5802/pmb.57 (https://doi.org/10.5802/pmb.57).DOI
  30. O. Ramaré and R. Rumely, Primes in arithmetic progressions , Math. Comp. 65 (1996), no. 213, 397--425. doi:10.1090/S0025-5718-96-00669-2 (https://doi.org/10.1090/S0025-5718-96-00669-2).DOI
  31. S. Ramanujan, A proof of Bertrand's postulate , J. Indian Math. Soc. 11 (1919), 181--182. https://ramanujan.sirinudi.org/Volumes/published/ram24.html.link
  32. W. Sawin, M. Shusterman, and M. Stoll, Irreducibility of polynomials with a large gap , Acta Arith. 192 (2020), no. 2, 111--139. doi:10.4064/aa180526-12-6 (https://doi.org/10.4064/aa180526-12-6).DOI
  33. A. Schinzel, Solution d'un problème de K. Zarankiewicz sur les suites de puissances consécutives de nombres irrationnels , Colloq. Math. 9 (1962), no. 2, 291--296. doi:10.4064/cm-9-2-291-296 (https://doi.org/10.4064/cm-9-2-291-296); correction, 12 (1964), 289. doi:10.4064/cm-12-2-289 (https://doi.org/10.4064/cm-12-2-289).DOI
  34. I. Schur, Gleichungen ohne Affekt , Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl. (1930), 443--449; reprinted in Gesammelte Abhandlungen , vol. III, Springer-Verlag, Berlin, 1973, 191--197.
  35. S. Seki, Irreducibility of truncations of the Catalan generating function , preprint (2026), arXiv:2609.10526v1 (https://arxiv.org/abs/2609.10526v1).arXiv
  36. J.-P. Serre, Topics in Galois Theory , second ed., notes written by H. Darmon, Research Notes in Mathematics, vol. 1, A K Peters, Wellesley, MA, 2008. doi:10.1201/b10588 (https://doi.org/10.1201/b10588).DOI
  37. K. M. Shokri, J. Shaffaf, and R. Taleb, Galois groups of Taylor polynomials of some elementary functions , Int. J. Number Theory 15 (2019), 1127--1141. doi:10.1142/S1793042119500623 (https://doi.org/10.1142/S1793042119500623).DOI
  38. P. Stevenhagen and H. W. Lenstra, Jr., Chebotarëv and his density theorem , Math. Intelligencer 18 (1996), no. 2, 26--37. doi:10.1007/BF03027290 (https://doi.org/10.1007/BF03027290).DOI
  39. Z.-W. Sun, entry A224416, The On-Line Encyclopedia of Integer Sequences , created 6 April 2013, consulted 24 October 2026. https://oeis.org/A224416.link
  40. Z.-W. Sun, comment dated 23 March 2013 on entry A000984, The On-Line Encyclopedia of Integer Sequences , accessed 24 October 2026. https://oeis.org/A000984.link
  41. D. Wei, Prime witnesses and eventual alternating--symmetric Galois groups in a Jacobi family , Zenodo preprint (2026). doi:10.5281/zenodo.23115023 (https://doi.org/10.5281/zenodo.23115023).DOI
  42. S. Weinbaum, Galois groups of shifted exponential polynomials , preprint, 2026, arXiv:2610.26873v2 (https://arxiv.org/abs/2610.26873v2).arXiv

Version history

  1. v1 · 48 pagesSubmitted by Khai-Hoan Nguyen-DangInitial depositSupersededOct 24, 2026
  2. v2 · 49 pagesSubmitted by Khai-Hoan Nguyen-DangSubmitted as a new versionSupersededOct 26, 2026
  3. v3 · 52 pagesSubmitted by Khai-Hoan Nguyen-DangSubmitted as a new versionCurrentOct 28, 2026