  In the context of Hilbert's tenth problem, an outstanding open case is that
of complex entire functions in one variable. A negative solution is known for
polynomials (by Denef) and for exponential polynomials of finite order (by
Chompitaki, Garcia-Fritz, Pasten, Pheidas, and Vidaux), but no other case is
known for rings of complex entire functions in one variable. We prove a
negative solution to the analogue of Hilbert's tenth problem for rings of
complex entire functions of finite order having lacunary power series expansion
at the origin.
