Polynomial Boundedness of Definably Complete Restricted Exponential Fields

Definitions and mathematical statement

Let $\mathcal K=(K;0,1,+,-,\cdot,<,\ldots)$ be an expansion of an ordered field. In the definitions below, "definable" means definable in $\mathcal K$ with parameters from $K$. No Archimedean assumption is imposed.

The structure $\mathcal K$ is definably complete if every nonempty definable subset $A\subseteq K$ that is bounded above has a least upper bound in $K$. It is polynomially bounded if, for every definable function $f:K\to K$, there exist a standard positive integer $n\in\mathbb Z_{\geq 1}$ and an element $b\in K$, with $b>1$, such that \[ \forall x\in K\quad \left(x>b\ \Longrightarrow\ |f(x)|\leq x^n\right). \] The integer $n$ and threshold $b$ may depend on $f$.

Source terminology requiring clarification. The supplied record does not define a "restricted exponential field": it gives neither its precise language nor the domain, normalization, and axioms of its restricted exponential operation. Let $\mathscr C_{\mathrm{src}}$ denote the class intended by that term in the cited report. This notation retains the source's intended class without choosing additional axioms. The parameter convention above is the usual convention for the two model-theoretic properties; its agreement with the report's precise language and conventions requires review.

Question (as supplied). Is every definably complete restricted exponential field polynomially bounded? With the preceding terminology, the proposed assertion is \[ \forall\mathcal K\in\mathscr C_{\mathrm{src}}\quad \left( \mathcal K\text{ is definably complete} \ \Longrightarrow\ \mathcal K\text{ is polynomially bounded} \right). \] A fully axiomatized formulation requires the missing definition of $\mathscr C_{\mathrm{src}}$. The supplied dated assessment describes this unconditional question as open and reports a result under a Schanuel-type hypothesis. That hypothesis and the exact conditional result are not specified in the record and are not added as assumptions to this question.

Short English statement

The question asks whether definable completeness forces every definable unary function in a restricted exponential field to be eventually bounded in absolute value by an integer power of its argument. The precise restricted exponential structure must be recovered from the cited source. The supplied open-status assessment and conditional-result description require independent verification.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30006585$, source problem code catalogue entry, title "Polynomial Boundedness of Definably Complete Restricted Exponential Fields." The record belongs to the Oberwolfach Reports Open Problems collection and assigns proposed year $2026$; no proposer is identified.
• Model Theory and Arithmetic (2026), Oberwolfach Reports, DOI 10.4171/owr/2026/3, pp. 157--216. Supplied source and literature reference "Oberwolfach Report 3/2026": https://doi.org/10.4171/owr/2026/3. The supplied excerpt presents the question as an open question; its speaker and exact location within the report are not identified.
• Supplied literature reference: On the elementary theory of the real exponential field, https://arxiv.org/abs/2603.08365. Authors and the precise supporting passage are not supplied.
• Literature assessment checked at 2026-08-22: the supplied curation labels the question open and states that the report describes polynomial boundedness as unknown in general, with current results only proving it under a Schanuel-type hypothesis. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and that OCR, notation, and source-dependent wording were repaired while retaining the complete target. These are supplied source-verification claims, not verification performed for this draft.