Polynomial Stability of Hypercyclic Entire Functions

Definitions and mathematical statement

Let $H({\mathbb{C}})$ be the space of entire functions with the topology of uniform convergence on compact sets, and let $D:H({\mathbb{C}})\to H({\mathbb{C}})$ be differentiation, $Df=f'$. Define \[ HC(D)=\{f\in H({\mathbb{C}}):\overline{\{D^nf:n\in{\mathbb{N}}_0\}}=H({\mathbb{C}})\}. \] The question is whether \[ \forall f\in HC(D)\ \forall p\in{\mathbb{C}}[z]\text{ nonconstant}:\qquad pf\in HC(D), \] where $(pf)(z)=p(z)f(z)$ is pointwise multiplication.

Equivalently, for every such $f,p$, every entire target $g$, every $R>0$, and every $\varepsilon>0$, must there exist $n\in{\mathbb{N}}_0$ such that \[ \sup_{|z|\le R}|(pf)^{(n)}(z)-g(z)|<\varepsilon\ ? \] The iterate may depend on all these data. The operator applied is differentiation of the product, not the polynomial differential operator $p(D)$ applied to $f$. No uniform iterate, growth bound, or genericity condition on $f$ is added to the universal question.

The supplied assessment reports a residual subclass of functions with the desired property. A residual set contains a countable intersection of open dense sets in the intended topology; that statement does not by itself settle all $f\in HC(D)$. The record does not say whether the reported subclass works for one fixed polynomial or simultaneously for all nonconstant polynomials, nor whether residuality is asserted in $H({\mathbb{C}})$ or in $HC(D)$. Those are partial-result qualifications requiring source review, not replacements for the full quantifiers above.

Short English statement

Must multiplication by every nonconstant polynomial preserve every entire function with a dense differentiation orbit? The question concerns derivatives of the product and all hypercyclic functions, beyond a reported residual subclass.

Sources

• Local UnsolvedMath registry, permanent ID 30000921, code catalogue entry, "Polynomial Stability of Hypercyclic Entire Functions." Proposed year 2008; proposer unspecified. The assessment calls these MacLane hypercyclic functions.
• Mini-Workshop: Complex Approximation and Universality (2008), Oberwolfach Reports, pp. 297--346, DOI 10.4171/owr/2008/06; https://doi.org/10.4171/owr/2008/06. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says the complete question was retained and function-space/membership notation restored.
• Literature triage checked 2026-08-21: open. It says polynomial multiplication is explicitly unresolved and a residual subclass is known to have the property, without precise polynomial quantifiers or ambient residuality. Created 2026-08-25; updated 2026-09-16. Exact partial-result scope and current status require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.