Hilbert's tenth problem for lacunary entire functions of finite order

TL;DR

This paper proves a negative solution to Hilbert's tenth problem for rings of lacunary entire functions of finite order, extending known results from polynomials and exponential polynomials.

Contribution

It establishes the first negative result for Hilbert's tenth problem in the context of lacunary entire functions of finite order.

Findings

Negative solution for rings of lacunary entire functions of finite order

Extends known results from polynomials and exponential polynomials

Addresses an open case in complex entire functions

Abstract

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.