  One of the main open problems regarding decidability of the existential
theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring
of entire holomorphic functions in one variable. In the direction of a negative
solution, we prove unsolvability of HTP for rings of exponential polynomials.
This provides the first known case of HTP for a ring of entire holomorphic
functions in one variable strictly containing the polynomials. The technique of
proof consists of an interaction between Arithmetic, Analysis, Logic, and
Functional Transcendence.
