The Diophantine problem for rings of exponential polynomials

A truncated second main theorem for algebraic tori with moving targets and applications

“…From the view points of studying the algebraic structure of the exponential polynomials, it basically says that the radical of an exponential polynomial is also an exponential polynomial. We refer to [10], [7], and [8] for other expositions in this direction; and [2] for related results in logic. The following is a natural question on the integrality of the ring of exponential polynomials over the ring of entire functions.…”

“…Then we have Δ = ∏ 𝑖<𝑗 (𝛼 𝑖 − 𝛼 𝑗 )2 , which is not zero since 𝐹 is irreducible. be the discriminant of 𝐻 with respect to 𝑌, we note that the entire function 𝐺 is equal to ∏ 1<𝑖<𝑗⩽𝑑 (𝛼 𝑖 − 𝛼 𝑗 ) 2 , which is nonzero.…”

Towards Hilbert’s tenth problem for rings of integers through Iwasawa theory and Heegner points

“…We remark that every exponential polynomial of finite order is holonomic, see Section 2. Thus, Theorem 1.1 (when A = C[z]) gives a negative solution to the analogue of Hilbert's tenth problem in cases that are completely different to those covered in [1].…”

“…The current progress has been much more difficult for subrings of H containing C[z], seen as L z -structures. A negative solution of the analogue of Hilbert's tenth problem was first established for complex polynomials by Denef [3] and, more recently, for exponential polynomials of finite order by Chompitaki, Garcia-Fritz, Pasten, Pheidas, and Vidaux [1]. This covers all known cases for subrings of…”

“…We thank Xavier Vidaux for several comments on a first version of this manuscript. This research was heavily influenced by the ideas and work of Thanases Pheidas, who was one of the driving forces in the study of extensions of Hilbert's tenth problem, especially in the case of rings and fields of functions; see for instance [1,6,8,9,10,11,12] among other works. We respectfully dedicate this article to his memory and legacy.…”