Hilbert's tenth problem for rings of holomorphic functions of bounded order

TL;DR

This paper addresses a key open problem by showing that Hilbert's tenth problem has no solution in rings of complex entire functions of bounded growth order, advancing understanding in number theory and complex analysis.

Contribution

It provides a negative solution to the analogue of Hilbert's tenth problem for rings of entire functions with bounded growth order, a significant step forward in the field.

Findings

Negative solution for all growth orders $ ho \,\geq 0$

Extends the understanding of Diophantine problems in complex function rings

Progress towards resolving the classical open problem

Abstract

One of the main open problems in the context of extensions of Hilbert's tenth problem (HTP) is the case of the ring of complex entire functions in one variable. Our main result provides a step towards an answer: For every $\rho\ge 0$, we give a negative solution to the analogue of HTP in the ring of complex entire functions in one variable of growth order at most $\rho$.