Skip to main content
SpringerLink
Log in

Vectors of type II Hermite–Padé approximations and a new linear independence criterion

  • Published:
Annali di Matematica Pura ed Applicata (1923 -) Aims and scope Submit manuscript

Abstract

We propose a linear independence criterion, and outline an application of it. Down to its simplest case, it aims at solving this problem: given three real numbers, typically as special values of analytic functions, how to prove that the \(\mathbb {Q}\)-vector space spanned by 1 and those three numbers has dimension at least 3, whenever we are unable to achieve full linear independence, by using simultaneous approximations, i.e. those usually arising from Hermite–Padé approximations of type II and their suitable generalizations. It should be recalled that approximations of type I and II are related, at least in principle: when the numerical application consists in specializing actual functional constructions of the two types, they can be obtained, one from the other, as explained in a well-known paper by Mahler (1968) Compos Math 19: 95–166. That relation is reflected in a relation between the asymptotic behavior of the approximations at the infinite place of \(\mathbb {Q}\). Rather interestingly, the two view-points split away regarding the asymptotic behaviors at finite places (i.e. primes) of \(\mathbb {Q}\), and this makes the use of type II more convenient for particular purposes. In addition, sometimes we know type II approximations to a given set of functions, for which type I approximations are not known explicitly. Our approach can be regarded as a dual version of the standard linear independence criterion, which essentially goes back to Siegel.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agarwal, R.P.: Difference equations and inequalities: theory, methods, and applications. 2nd, revised and expanded ed., Pure and Applied Mathematics, Marcel Dekker. 228. New York, xiii+971 (2000)

  2. Aitken, A.C.: The normal form of compound and induced matrices. Proc. Lond. Math. Soc. 2(38), 354–376 (1934)

    MathSciNet  MATH  Google Scholar 

  3. Amoroso, F.: Indépendance linéaire décalée, manuscript (2004)

  4. Bedulev, E.V.: On the linear independence of numbers over number fields. Mat. Zametki [Math. Notes] 64(44), 506–517 (1998). ([440–449])

  5. Bombieri, E., Vaaler, J.: On Siegel’s lemma. Invent. Math. 73(1983), 11–32 (1983);addendum: ibid. 75 (1984), 377

  6. Bundschuh, P., Töpfer, T.: Über lineare Unabhängigkeit. Monatsh. Math. 117(1–2), 17–32 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Buslaev, V.I.: Relations for the coefficients and singular points of a function. Mat. Sb. 131(173), (1986), 357–384 (Russian); English: Math. USSR Sb. 59(2), 349–377 (1988)

  8. Casorati, F.: Il calcolo delle differenze finite interpretato ed accresciuto di nuovi teoremi a sussidio principalmente delle odierne ricerche basate sulla variabilità complessa. Ann. Mat. Pura Appl. (2) 10, 10–43 (1880)

    Article  MATH  Google Scholar 

  9. Chantanasiri, A.: On the criteria for linear independence of Nesterenko, Fischler and Zudilin. Chamchuri J. Math. 2(1), 31–46 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Chantanasiri, A.: Généralisation des critères pour l’indépendance linéaire de Nesterenko, Amoroso, Colmez, Fischler et Zudilin. Ann. Math. Blaise Pascal 19(1), 75–105 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chudnovsky, D.V., Chudnovsky, G.V.: Applications of Padé approximations to diophantine inequalities in values of G-functions. In: Chudnovsky D.V., Chudnovsky G.V., Cohn H., Nathanson M.B. (eds.), Number Theory. Lecture Notes in Mathematics 1135, Springer, Berlin, pp. 9–51 (1985)

  12. Colmez, P.: Arithmétique de la fonction zêta, in: La fonction zêta. Berline, N., Sabbah, C. (ed). Palaiseau: Les Éditions de l’École Polytechnique, pp. 37–164 (2003)

  13. Coppel, W.A.: Disconjugacy, Lecture Notes in Mathematics. 220. Springer, Berlin, p. 147 (1971)

  14. Dauguet, S.: Généralisations quantitatives du critère d’indépendance linéaire de Nesterenko. J. Théor. Nombres Bordeaux 27(2), 483–498 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. David, S., Hirata-Kohno, N., Kawashima, M.: Can polylogarithms at algebraic points be linearly independent?, arXiv: 1912.03811 [math.NT], Mosc. J. Comb. Number Theory 9:4 (2020), 389–406

  16. Dubickas, A.: On the approximation of \(\pi /\sqrt{3}\) by rational fractions Vestn. Mosk. Univ., Ser. I 1987, No. 6 (1987), 73–76 (Russian); English: Mosc. Univ. Math. Bull. 42(6), 76–79 (1987)

  17. Fischler, S.: Nesterenko’s criterion when the small linear forms oscillate. Arch. Math. 98(2), 143–151 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fischler, S.: Nesterenko’s linear independence criterion for vectors. Monatsh. Math. 177(3), 397–419 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fischler, S., Hussain, M., Kristensen, S., Levesley, J.: A converse to linear independence criteria, valid almost everywhere. Ramanujan J. 38(3), 513–528 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fischler, S., Rivoal, T.: Multiple zeta values, Padé approximation and Vasilyev’s conjecture. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 15(15), 24–1 (2016)

    MATH  Google Scholar 

  21. Fischler, S., Rivoal, T.: Linear independence of values of G-functions, II. Outside the disk of convergence, arXiv: 1811.08758 [math.NT], HAL: hal-01927576, Ann. Math. Qué. (2020)

  22. Fischler, S., Zudilin, W.: A refinement of Nesterenko’s linear independence criterion with applications to zeta values. Math. Ann. 347(4), 739–763 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Flanders, H.: A note on the Sylvester-Franke theorem. Am. Math. Monthly 60(8), 543–545 (1953)

    MathSciNet  MATH  Google Scholar 

  24. Franke, E.: Ueber Determinanten aus Unterdeterminanten. J. Reine Angew. Math. 61, 350–355 (1863)

    MathSciNet  Google Scholar 

  25. Goncharov, A.B.: Multiple polylogarithm, cyclotomy and modular complexes. Math. Res. Lett. 5, 497–516 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hartman, P.: Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity. Trans. Am. Math. Soc. 246, 1–30 (1978)

    MathSciNet  MATH  Google Scholar 

  27. Hata, M.: On the linear independence of the values of polylogarithmic functions. J. Math. Pures Appl. (9) 69(2), 133–173 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Hata, M.: Rational approximations to \(\pi \) and some other numbers. Acta Arith. 63(4), 335–349 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hata, M.: Rational approximations to the dilogarithm. Trans. Am. Math. Soc. 336(1), 363–387 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hata, M.: \(\mathbb{C}^2\)-saddle method and Beukers’ integral, ibid. 352 (2000), 4557–4583

  31. Krattenthaler, C.: Advanced determinant calculus. Sém. Lothar. Combin. 42, B42q, 67pp (1999)

  32. Laurent, M., Roy, D.: Criteria of algebraic independence with multiplicities and approximation by surfaces. J. Reine Angew. Math. 536, 65–114 (2001)

    MathSciNet  MATH  Google Scholar 

  33. Littlewood, D.E.: On induced and compound matrices. Proc. Lond. Math. Soc. 2(40), 370–381 (1935)

    MathSciNet  MATH  Google Scholar 

  34. Mahler, K.: Perfect systems. Compos. Math. 19, 95–166 (1968)

    MathSciNet  MATH  Google Scholar 

  35. Marcovecchio, R.: Linear independence of linear forms in polylogarithms. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5, 1–11 (2006)

    MathSciNet  MATH  Google Scholar 

  36. Marcovecchio, R.: Multiple Legendre polynomials in diophantine approximation. Int. J. Number Theory 10, 1829–1855 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Marcovecchio, R.: Linear independence of polylogarithms at algebraic points. Mosc. J. Comb. Number Theory 6(2–3), 208–232 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Meray, C.: Extension aux équations simultanées des formules de Newton pour le calcul des sommes de puissances semblables des racines des équations entières. Ann. Sci. École Norm. Sup. 1(4), 159–193 (1867)

    Article  MathSciNet  Google Scholar 

  39. Nesterenko, Yu.V.: On the linear independence of numbers, Vestn. Mosk. Univ. Ser. I, No. 1 [Mosc. Univ. Math. Bull. 40:1] (1985), 46–49 [69–74]

  40. Nesterenko, Yu.V.: On a criterion of linear independence of p-adic numbers. Manuscripta Math. 139(3–4), 405–414 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nikishin, E.M.: On logarithms of natural numbers, Izv. Akad. Nauk SSSR Ser. Mat. 43:6 (1979), 1319–1327;correction: ibid. 44:4 (1980), 972 [Math. USSR-Izv. 15:3 (1980), 523–530]

  42. The PARI Group, PARI/GP version 2.9.4, Univ. Bordeaux, 2017, http://pari.math.u-bordeaux.fr/

  43. Perron, O.: Über die Poincarésche lineare Differenzengleichung. J. Reine Angew. Math. 137, 6–64 (1909)

    MATH  Google Scholar 

  44. Perron, O.: Über Summengleichungen und Poincaresche Differenzengleichungen. Math. Ann. 84, 1–15 (1921)

    Article  MathSciNet  MATH  Google Scholar 

  45. Philippon, P.: Critères pour l’indépendance algébrique. Publ. Math. Inst. Hautes Études Sci. 64, 5–52 (1986)

    Article  MATH  Google Scholar 

  46. Pincherle, S., Amaldi, U.: Le operazioni distributive e le loro applicazioni all’analisi, Bologna, Zanichelli, xii+490 (1901), reprinted at the occasion of the XIX Congress of the U.M.I, Bologna, September 12-17, 2011

  47. Pinna, F., Viola, C.: The saddle-point method in \(\mathbb{C}^N\) and the generalized Airy functions. Bull. Soc. Math. France 147(2), 221–257 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  48. Pituk, M.: More on Poincaré’s and Perron’s theorems for difference equations. J. Diff. Equ. Appl. 8(3), 201–216 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  49. Pituk, M.: A link between the Perron-Frobenius theorem and Perron’s theorem for difference equations. Linear Algebra Appl. 434, 490–500 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Poincaré, H.: Sur les équations lineaires aux différentielles ordinaires et aux différences finies. Am. J. Math. 7, 203–258 (1885)

    Article  MATH  Google Scholar 

  51. Price, G.B.: Some identities in the theory of determinants. Am. Math. Monthly 54(2), 75–90 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  52. Rhin, G., Toffin, P.: Approximants de Padé simultanés de logarithmes. J. Number Theory 24, 284–297 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rhin, G., Viola, C.: On a permutation group related to \(\zeta (2)\). Acta Arith. 77(1), 23–56 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rhin, G., Viola, C.: Linear independence of 1, Li1 and Li2. Mosc. J. Comb. Number Theory 8(1), 81–96 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  55. Rukhadze, E..A.: A lower bound for the approximations of \(\mathrm{ln}2\) by rational numbers. Vestn. Mosk. Univ. Ser. I Mat. Mekh. 6, (1987), 25–29 (Russian); English: Mosc. Univ. Math. Bull. 42(66), 30–35 (1987)

  56. Sylvester, J.J.: On the relations between the minor determinants of linearly equivalent quadratic functions. Phil. Mag. (4) 1(4), 395–405 (1851)

    Google Scholar 

  57. Töpfer, T.: Über lineare Unabhängigkeit in algebraischen Zahlkörpern. Result. Math. 25(1–2), 139–152 (1994)

    Article  MATH  Google Scholar 

  58. Töpfer, T.: An axiomatization of Nesterenko’s method and applications on Mahler functions. J. Number Theory 49(1), 1–26 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  59. Töpfer, T.: An axiomatization of Nesterenko’s method and applications on Mahler functions. II. Compos. Math. 95(3), 323–342 (1995)

    MathSciNet  MATH  Google Scholar 

  60. Tornheim, L.: The Sylvester-Franke Theorem. Am. Math. Monthly 59(6), 389–391 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  61. Viola, C.: On Siegel’s method in diophantine approximation to transcendental numbers. Rend. Semin. Mat. Univ. Politec. Torino 53(4), 455–469 (1995)

    MathSciNet  MATH  Google Scholar 

  62. Viola, C., Zudilin, W.: Linear independence of dilogarithmic values. J. Reine Angew. Math. 736, 193–223 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  63. Waldschmidt, M.: Introduction to Diophantine methods: irrationality and transcendence, notes of the course, Ho Chi Minh University of Natural Sciences, September 12–October 4, 2007. Mission effectuée dans le cadre du PICS Formath Vietnam, 94 pp. (https://webusers.imj-prg.fr/~michel.waldschmidt/coursHCMUNS2007.html, last update: 03/04/2019)

  64. Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q’’) multi-sum/integral. Invent. Math. 108(3), 575–633 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  65. Zudilin, V.V.: Difference equations and the irrationality measure of numbers. Tr. Mat. Inst. Steklova, Anal. Teor. Chisel i Prilozh. [Proc. Steklov Inst. Math.] 218, 165–178 (1997). ([160–174])

  66. Zudilin, W.: Two hypergeometric tales and a new irrationality measure of \(\zeta (2)\). Ann. Math. Qué. 38, 101–117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  67. Zudilin, W.: A determinantal approach to irrationality. Constr. Approx. 45(2), 301–310 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The core of the present note, a short less-than-seven-pages draft without applications, was written several years ago partly during, partly after a stay at the Centre International de Rencontres Mathématiques de Luminy, France. However, my attention to the topic was recently refreshed by the papers [15] and [62], and specially by the proof of [62, Lemma 5.3]. Along the time, I had the pleasure of chatting on this topic with F.Amoroso, M.Laurent and W.Zudilin; a special thank to them, and to whom else made all this possible, in a way or another. I owe a debt of gratitude to an anonymous referee for a very careful reading. Some computations in Sect. 4 were made with the help of the free software Pari/GP [42].

Author information

Authors and Affiliations

  1. Università degli Studi Gabriele d′Annunzio di Chieti-Pescara Dipartimento di Ingegneria e Geologia, Pescara, Italy

    Raffaele Marcovecchio

Authors
  1. Raffaele Marcovecchio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raffaele Marcovecchio.

Additional information

Dedicated to Wadim Zudilin, with warm wishes, on the occasion of his 50th birthday.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marcovecchio, R. Vectors of type II Hermite–Padé approximations and a new linear independence criterion. Annali di Matematica 200, 2829–2861 (2021). https://doi.org/10.1007/s10231-021-01104-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10231-021-01104-7

Keywords

Mathematics Subject Classification

Search

Navigation