On polynomial growth functions of D0L-systems

The aim of this paper is to prove that every polynomial function that maps the natural integers to the positive integers is the growth function of some D0L-system.

💡 Research Summary

The paper addresses a fundamental question in the theory of deterministic context‑free L‑systems (D0L‑systems): which functions can appear as the growth function of a D0L‑system, i.e., the length of the word after n derivation steps? While previous work has largely focused on linear and exponential growth, the existence of D0L‑systems whose growth follows an arbitrary polynomial with positive integer coefficients had remained open. The authors close this gap by proving that every such polynomial can be realized as the growth function of a suitably constructed D0L‑system.

The authors begin by formalising the notion of a D0L‑system (\mathcal{S} = (\Sigma, \varphi, w_0)) where (\Sigma) is a finite alphabet, (\varphi) a deterministic morphism, and (w_0) the axiom. The growth function is defined as (g_{\mathcal{S}}(n) = | \varphi^n(w_0) |). They then fix an arbitrary polynomial \