A conjecture by Leon Ehrenpreis about zeroes of exponential polynomials

Alain Yger (1)

1 IMB - Institut de Mathématiques de Bordeaux

Leon Ehrenpreis proposed in his 1970 monograph Fourier Analysis in sev- eral complex variables the following conjecture : the zeroes of an exponential polynomial PM 0 bk(z)ei kz, bk 2 Q[X], k 2 Q\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 di erential delayed operators in n variables with algebraic constant coe cients, as well as algebraic delays, is closed or not in the Paley-Wiener algebra b E(Rn). In this survey, I present various analytic approaches to such a question, involving either the Schanuel-Ax formal conjecture, or 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 where introduced by Y. Andr e to revisit the Lindemann-Weierstrass theorem, could also be understood as an indication for rigidity constraints for example in Ritt's factorisation 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.

Mots clés: exponential-polynomials; spectral synthesis; ordinary differential equations

Domaines: Variables complexes [math.CV]; Théorie des nombres [math.NT]

Alain Yger. A conjecture by Leon Ehrenpreis about zeroes of exponential polynomials. Farkas, H.M, Gunning, R.C, Knopp, M.I, Taylor, B.A. From Fourier Analysis and Number Theory to Radon Transform and Geometry, Springer-Verlag, pp.517-535, 2013, Developpments in Mathematics, 28, 978-1-4614-4075-8. ⟨hal-00618147v2⟩