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.
This follows from Theorem 11.2 in [36].
- 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.
Here again, additional arithmetic information on F does not impose any arithmetic constraint on γ.
- 4.
Certainly, the coefficients and frequencies of such an exponential polynomial f are in \(\overline{\mathbb{Q}}\).
- 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.
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.
See, for example, [33], Chap. VIII, for a pedestrian presentation and a proof.
- 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.
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.
- 11.
That is on concepts of algebraic, not really arithmetic, nature, though arithmetics is deeply involved.
- 12.
Polya’s theory, see also [7].
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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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