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A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials

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Book cover From Fourier Analysis and Number Theory to Radon Transforms and Geometry

Part of the book series: Developments in Mathematics ((DEVM,volume 28))

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Abstract

Leon Ehrenpreis proposed in his 1970 monograph Fourier Analysis in several complex variables the following conjecture: the zeroes of an exponential polynomial \({\sum \nolimits }_{0}^{M}{b}_{k}(z){\mathrm{e}}^{\mathrm{i}{\alpha }_{k}z}\), \({b}_{k} \in \overline{\mathbb{Q}}[X]\), \({\alpha }_{k} \in \overline{\mathbb{Q}} \cap \mathbb{R}\) are well separated with respect to the Paley–Wiener weight. Such a conjecture remains essentially open (besides some very peculiar situations). But it motivated various analytic developments carried by C.A. Berenstein and the author, in relation with the problem of deciding whether an ideal generated by Fourier transforms of differential delayed operators in n variables with algebraic constant coefficients, as well as algebraic delays, is closed or not in the Paley–Wiener algebra \(\widehat{\mathcal{E}}({\mathbb{R}}^{n})\). In this survey, I present various analytic approaches to such a question, involving either the Schanuel-Ax formal conjecture or \(\mathcal{D}\)-modules technics based on the use of Bernstein–Sato relations for several functions. Nevertheless, such methods fail to take into account the intrinsic rigidity which arises from arithmetic hypothesis: this is the reason why I also focus on the fact that Gevrey arithmetic methods, that were introduced by Y. André to revisit the Lindemann–Weierstrass theorem, could also be understood as an indication for rigidity constraints, for example, in Ritt’s factorization theorem of exponential sums in one variable. The objective of this survey is to present the state of the art with respect to L. Ehrenpreis’s conjecture, as well as to suggest how methods from transcendental number theory could be combined with analytic ideas, in order precisely to take into account such rigidity constraints inherent to arithmetics.

Mathematics Subject Classification (2010): 42A75 (Primary), 65Q10, 11K60, 11J81, 39A70, 14F10 (Secondary)

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Notes

  1. 1.

    This follows from Theorem 11.2 in [36].

  2. 2.

    Unfortunately, even when one specifies arithmetic conditions on the ideal I, such as the generating polynomials have algebraic coefficients, nothing more precise can be asserted about the constant γ. Indeed, this is the main stumbling block to such a result being an efficient tool in proving Conjecture 1.2 or even Conjecture 1.4.

  3. 3.

    Here again, additional arithmetic information on F does not impose any arithmetic constraint on γ.

  4. 4.

    Certainly, the coefficients and frequencies of such an exponential polynomial f are in \(\overline{\mathbb{Q}}\).

  5. 5.

    That is, of course, is identically zero. Nevertheless, it seems better to keep this formulation to view the statement as the effect of arithmetic rigidity constraints in Ritt’s factorization theorem.

  6. 6.

    More generally, one may replace \(\mathbb{K}(X)\) by some unitary \(\mathbb{K}\)-algebra containing \(\mathbb{K}(X)\), such as \(\mathbb{K}\,[[X]]\), and introduce then the notion of Xd ∕ dX-module of finite type over \(\mathbb{K}\,[[X]]\).

  7. 7.

    See, for example, [33], Chap. VIII, for a pedestrian presentation and a proof.

  8. 8.

    To say it briefly, a G-function is a formal power series in \(\overline{\mathbb{Q}}\,[[X]]\) which is in the kernel of some element in \(\overline{\mathbb{Q}}[X,d/dX]\) and, at the same time, has a finite logarithmic height, when considered as a power series in \(\overline{\mathbb{Q}}\,[[X]]\) (see [3] for the notion of logarithmic height for a power series).

  9. 9.

    The lines which follow intend just to sketch what could be a conjectural approach to Conjecture 1.4 for exponential sums f such that Γ(f) has small rank.

  10. 10.

    Note that this work of L. Ehrenpreis appeared in the Lecture Notes volume where appeared also the important results by Chudnovsky [29, 30].

  11. 11.

    That is on concepts of algebraic, not really arithmetic, nature, though arithmetics is deeply involved.

  12. 12.

    Polya’s theory, see also [7].

References

  1. M. Andersson, Residue currents and ideals of holomorphic functions, Bull. Sci. Math. 128 (2004), no. 6, pp. 481–512.

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Andersson, E. Wulcan, Residue currents with prescribed annihilator ideals, Ann. Sci. École Norm. Sup. 40 (2007), pp. 985–1007.

    MATH  MathSciNet  Google Scholar 

  3. Y. André, G-Functions and Geometry, Aspects of Mathematics, Vieweg, 1989.

    MATH  Google Scholar 

  4. Y. André, Séries Gevrey de type arithmétique, I. Théorèmes de pureté et de dualité, Annals of Mathematics 151 (2000), pp. 705–740.

    MATH  Google Scholar 

  5. Y. André, Séries Gevrey de type arithmétique, II. Transcendance sans transcendance, Annals of Mathematics 151 (2000), pp. 741–756.

    MATH  Google Scholar 

  6. J. Ax, On Schanuel’s conjectures, Ann. of Math. 93 (1971), pp. 252–268.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Bellman, K. Cooke, Differential difference equations, Academic Press, New York, 1963.

    MATH  Google Scholar 

  8. C. A. Berenstein, A. Dostal, The Ritt theorem in several variables, Ark. Math. 12 (1974), pp. 267–280.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. Baker, Transcendental Number Theory, Cambridge University Press, 1979.

    Google Scholar 

  10. A. Baker, G. Wülstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), pp. 19–62.

    MATH  MathSciNet  Google Scholar 

  11. R. Bahloul, Global generic Bernstein–Sato polynomial on an irreducible algebraic scheme, Proc. Japan Acad. 79, Ser. A (2003), pp. 146–149.

    Google Scholar 

  12. F. Battaglia, E. Prato, Generalized toric varieties for simple non rational convex polytopes, Intern. Math. Res. Notices 24 (2001), pp. 1315–1337.

    Article  MathSciNet  Google Scholar 

  13. C. A. Berenstein, R. Gay, Complex analysis and special topics in harmonic analysis, Springer-Verlag, New York, 1995.

    Book  MATH  Google Scholar 

  14. D. Brownawell, Pairs of polynomials small at a number to certain algebraic numbers, Séminaire Delange-Pisot-Poitou 11, 1975-1976, pp. 1–12.

    Google Scholar 

  15. C. A. Berenstein, A. Yger, On Lojasiewicz type inequalities for exponential polynomials, J. Math. Anal. Appl. 129 (1988), pp. 166–195.

    Article  MATH  MathSciNet  Google Scholar 

  16. C. A. Berenstein, A. Yger, Ideals generated by exponential polynomials, Advances in Mathematics 60 (1986), pp. 1–80.

    Article  MATH  MathSciNet  Google Scholar 

  17. C. A. Berenstein, A. Yger, Exponential polynomials and \(\mathcal{D}\)-modules, Compositio Mathematica 95 (1995), pp. 131–181.

    MATH  MathSciNet  Google Scholar 

  18. C. A. Berenstein, A. Yger, Division interpolation methods and Nullstellensätze, pp. 41–59 in Analysis, Geometry, Number Theory: The Mathematics of Leon Ehrenpreis, E. L. Grinberg [et al.] (eds.), Contemporary Mathematics 251, AMS 1999.

    Google Scholar 

  19. C. A. Berenstein, R. Gay, A. Vidras, A. Yger, Residue currents and Bézout identities, Progress in Mathematics 114, Birkhäuser, 1993.

    Google Scholar 

  20. C. A. Berenstein, A. Vidras, A. Yger, Multidimensional residue theory and applications, manuscript in preparation.

    Google Scholar 

  21. D. Bertrand, On André’s proof of the Siegel-Shidlovsky theorem, pp. 51–63 in Colloque Franco-Japonais: Théorie des Nombres Transcendants (Tokyo, 1998), Sem. Math. Sci., 27, Keio Univ., Yokohama, 1999.

    Google Scholar 

  22. D. Bertrand, Théories de Galois différentielles et transcendance, Annales Inst. Fourier 59 (2009), no. 7, pp. 2773–2803.

    Article  MATH  MathSciNet  Google Scholar 

  23. F. Beukers, J. P. Bézivin, P. Robba, An alternative proof of the Lindemann-Weierstrass theorem, American Monthly 97 (1990), no. 3, pp. 193–197.

    Article  MATH  Google Scholar 

  24. J. P. Bézivin, P. Robba, A new p-adic method for proving irrationality and transcendence results, Annals of Mathematics 129 (1989) pp. 151–160.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. E. Björk, Rings of of differential operators, North-Holland, Amsterdam, 1979.

    MATH  Google Scholar 

  26. J. E. Björk, Analytic \(\mathcal{D}\) -modules and applications, Mathematics and its Applications 247, Kluwer Academic Publishers Group, Dordrecht, 1993.

    Google Scholar 

  27. J. E. Björk, Residues and \(\mathcal{D}\)-modules, pp. 605–651 in The legacy of Niels Henrik Abel, O.A. Laudal, R. Piene (eds.), Springer-Verlag, Berlin (2004).

    Google Scholar 

  28. C. Chabauty, Sur les équations diophantiennes liées aux unités d’un corps de nombres algébrique fini, Ann. Math. Pura Appl. 17 (1938), pp. 127–168.

    Article  MathSciNet  Google Scholar 

  29. D. Chudnovsky, G. Chudnovsky, Applications of Padé approximations to diophantine inequalities in values of G-functions, pp. 9–51 in Number theory (New York 1983-1984), Lecture Notes in Math. 1135, Springer-Verlag, Berlin (1985).

    Google Scholar 

  30. D. Chudnovsky, G. Chudnovsky, Applications of Padé approximation to the Grothendieck conjecture on linear differential equations, pp. 52–100 in Number theory (New York 1983-1984), Lecture Notes in Math. 1135, Springer-Verlag, Berlin (1985).

    Google Scholar 

  31. R. Coleman, A generalization of the Ax-Schanuel theorem, Amer. J. Math. 102 (1980) pp. 595–624.

    Article  MATH  MathSciNet  Google Scholar 

  32. L. di Vizio, Sur la théorie géométrique des G-fonctions, le théorème de Chudnovsky à plusieurs variables, Math. Ann. 319 (2001), pp. 181–213.

    Article  MATH  MathSciNet  Google Scholar 

  33. B. Dwork, G. Gerotto, F. J. Sullivan, An introduction to G-functions, Annals of Mathematics Studies 133, Princeton University Press, 1994.

    Google Scholar 

  34. G. Diaz, Grands degrés de transcendance pour des familles d’exponentielles, C.R. Acad. SCi. Paris Sér. I Math. 305 (1987), no. 5, pp. 159–162.

    Google Scholar 

  35. F. Ehlers, The Weyl algebra, pp. 173–206 in Algebraic \(\mathcal{D}\) -modules (Perspectives in Mathematics), A. Borel, J. Coates, S. Helgason (eds.), Academic Press, Boston, 1987.

    Google Scholar 

  36. L. Ehrenpreis, Fourier Analysis in several complex variables, Wiley, 1970.

    Google Scholar 

  37. L. Ehrenpreis, Transcendental numbers and partial differential equations, pp. 112–125 in Number theory (New York, 1983-1984), Lecture Notes in Math. 1135, Springer-Verlag, Berlin (1985).

    Google Scholar 

  38. M. Forsberg, M. Passare, A. Tsikh, Laurent determinants and arrangements of hyperplane amoebas, Adv. in Maths. 151 (2000), pp. 45–70.

    Article  MATH  MathSciNet  Google Scholar 

  39. F. Gramain, Solutions indéfiniment dérivables et solutions presque périodiques d’une équation de convolution, Bull. Soc. Math. France 104 (1976), pp. 401–408.

    MATH  MathSciNet  Google Scholar 

  40. E. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math. 90, 1968, pp. 1151–1164.

    Article  MATH  MathSciNet  Google Scholar 

  41. D. Gurevich, Closed ideals with zero dimensional root set in certain rings of holomorphic functions, J. Soviet Math. 9 (1978), pp. 172–182.

    Article  MATH  Google Scholar 

  42. A. Gyoja, Bernstein–Sato’s polynomial for several analytic functions, J. Math. Kyoto Univ. 33, (1993), no. 2, pp. 399–411.

    MATH  MathSciNet  Google Scholar 

  43. A. Henriques, An analog of convexity for complements of amœbas of higher codimension, an answer to a question asked by B. Sturmfels, Adv. Geom. 4 (2004), no. 1, pp. 61–73.

    Google Scholar 

  44. N. Katz, Algebraic solutions of Differential Equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), pp. 1–118.

    Article  MATH  MathSciNet  Google Scholar 

  45. B. Ya. Kazarnovskii, On zeros of exponential sums, Dokl. Akad. Nauk SSSR, 257 (1981), no. 4, pp. 804–808.

    MathSciNet  Google Scholar 

  46. B. Ya. Kazarnovskii, Exponential analytic sets, Functional Analysis and its applications, 31 (1997), no. 2, pp. 86–94.

    MATH  Google Scholar 

  47. C. Moreno, The zeroes of exponential polynomials (I), Compositio Math. 26 (1973), no. 1, 69–78.

    MATH  MathSciNet  Google Scholar 

  48. M. Passare, H. Rullgard, Amoebas,Monge-Ampère measures, and triangulations of the Newton polytope, Duke Math. J. 121 (2004), no. 3, pp. 481–507.

    Article  MATH  MathSciNet  Google Scholar 

  49. K. Purbhoo, A Nullstellensatz for amoebas, Duke Mathematical Journal 141 (2008), no. 3, 407–445.

    Article  MATH  MathSciNet  Google Scholar 

  50. A. Yu. Rashkovskii, Zeros of holomorphic almost periodic mappings with independent components. Complex Variables 44 (2001) 299–316.

    Article  MATH  MathSciNet  Google Scholar 

  51. J. F. Ritt, A factorization theory for functions \({\sum \nolimits }_{i=1}^{n}{a}_{i}{e}^{{\alpha }_{i}x}\), Trans. Amer. Math. Soc. 29 (1927), no. 3, pp. 584–596.

    MATH  MathSciNet  Google Scholar 

  52. L. I. Ronkin, Functions of completely regular growth, Mathematics and its Applications (Soviet Series), 81. Kluwer Academic Publishers, Dordrecht, 1992.

    Google Scholar 

  53. L. I. Ronkin, On zeros of almost periodic functions generated by holomorphic functions in multicircular domain, pp. 243–256 in Complex Analysis in Modern Mathematics, Fazis, Moscow, 2001.

    Google Scholar 

  54. C. Sabbah, Proximité évanescente II. Équations fonctionnelles pour plusieurs fonctions analytiques, Compositio Math 64 (1987), pp. 213–241.

    MATH  MathSciNet  Google Scholar 

  55. J. Silipo, Amibes de sommes d’exponentielles, Canadian J. Math. 60, 1 (2008), pp. 222–240.

    Article  MATH  MathSciNet  Google Scholar 

  56. R. Tijdeman, A. Van der Poorten, On common zeroes of exponential polynomials, Enseignement Mathématique 21 (1975), no. 2, pp. 57–67.

    MATH  Google Scholar 

  57. A. Van der Poorten, A note on the zeroes of exponential polynomials, Compositio Math. 31 (1975), no. 2, pp. 109–113.

    MATH  MathSciNet  Google Scholar 

  58. M. Waldschmidt, Open diophantine problems, Mosc. Math. J. 4, no. 1 (2004), pp. 245–305.

    Google Scholar 

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  1. Institut de Mathématiques de Bordeaux, Université Bordeaux 1, Talence, France

    Alain Yger

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  1. , Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Hershel M. Farkas

  2. , Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, 08544, New Jersey, USA

    Robert C. Gunning

  3. , Department of Mathematics, Temple University, Wachman Hall 608, Philadelphia, 19122, Pennsylvania, USA

    Marvin I. Knopp

  4. , Department of Mathematics, University of Michigan, Church St. 530, Ann Arbor, 48109, Michigan, USA

    B. A. Taylor

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Dedicated to the memory of Leon Ehrenpreis

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Yger, A. (2013). A Conjecture by Leon Ehrenpreis About Zeroes of Exponential Polynomials. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_26

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