Unconditional completeness, polynomial boundedness, and residue-field models of the definably complete exponential theories $T^{dc}_{\exp}$ and $T^{dc}_{\mathrm{e}}$

The first-order theory $T_{\exp}$ of the real exponential field is known to be decidable assuming the real Schanuel conjecture $\mathrm{SC}_{\mathbb{R}}$ (Macintyre–Wilkie, 1996), and Berarducci and Gallinaro (2026) proved, again under $\mathrm{SC}_{\mathbb{R}}$, that $T_{\exp}$ is axiomatized by the recursive theory $T^{dc}_{\exp}$ of definably complete exponential fields satisfying $\exp'=\exp$; unconditionally they established the model completeness of the companion theory $T^{dc}_{\mathrm{e}}$ of definably complete restricted exponential fields. Berarducci and Gallinaro leave open three questions, which together ask for the unconditional model theory of these definably complete exponential theories: (1) is $T^{dc}_{\mathrm{e}}$, or at least $T^{dc}_{\exp}$, complete without any transcendence hypothesis — equivalently, can $T^{dc}_{\exp}=T_{\exp}$, and hence the decidability of $T_{\exp}$, be proved unconditionally? (2) is $T^{dc}_{\mathrm{e}}$ polynomially bounded, i.e. is every unary definable function in every model ultimately bounded by a polynomial? (3) is the induced restricted exponential field on the residue field $\mathcal{O}/\mathfrak{m}$ of a convex subring of a model of $T^{dc}_{\mathrm{e}}$ itself a model of $T^{dc}_{\mathrm{e}}$? No unconditional progress on any of the three questions is known; the polynomial-boundedness question is reported as open by Freni (2026). All three questions are open.

Background

The real exponential field $\mathbb{R}_{\exp}$ is the ordered field of real numbers expanded by the usual exponential function, studied in the first-order language of ordered rings with an added function symbol $\exp$; its complete first-order theory is denoted $T_{\exp}$. Tarski (1951) gave a decision method for the corresponding theory of pure real algebra, and determining the decidability and the axiomatization of $T_{\exp}$ has been a central open problem of the model theory of the reals ever since (Tarski's problem for $\mathbb{R}_{\exp}$). Wilkie (1989, 1996) proved that $T_{\exp}$ is model complete, meaning every embedding between models of $T_{\exp}$ is elementary; under model completeness every formula is equivalent, modulo the theory, to an existential formula (a quantifier-free formula preceded by a block of existential quantifiers), but this does not by itself decide which first-order sentences the theory proves. Macintyre and Wilkie (1996) proved that $T_{\exp}$ is decidable — there is an algorithm deciding membership in $T_{\exp}$ — assuming the real Schanuel conjecture $\mathrm{SC}_{\mathbb{R}}$: for every tuple $a=(a_1,\dots,a_n)$ of $\mathbb{Q}$-linearly independent real numbers, the tuple $(a_1,\dots,a_n,e^{a_1},\dots,e^{a_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$. They also showed that decidability of $T_{\exp}$ is equivalent to a weak form of Schanuel's conjecture (WSC), expressible as a first-order axiom scheme, so removing the transcendence hypothesis is not a matter of bookkeeping: unconditional decidability would itself prove WSC.

Berarducci and Gallinaro (2026) revisit this program using definable completeness, a weakening of the familiar tameness property of o-minimality. Recall that a structure expanding a dense linear order is o-minimal if every definable subset of the order is a finite union of points and open intervals, and definably complete if every definable subset with an upper bound has a least upper bound. Definable completeness was studied for its own sake by Fornasiero and Servi (2010), who showed that definably complete structures satisfy an analogue of the Baire category theorem, by Hieronymi (2013), who showed that this Baire property already follows from definable completeness alone, and by Fornasiero and Hieronymi (2015), who established a fundamental dichotomy for definably complete expansions of ordered fields. Berarducci and Gallinaro consider two theories with recursive first-order axiom schemes: $T^{dc}_{\exp}$ (their Definition 4.5), whose models are the definably complete expansions of an ordered field by a function $\exp$ with $\exp'(x)=\exp(x)$ for all $x$ and $\exp(0)=1$ — these models coincide with the EXP-fields of Krapp (2023) — and $T^{dc}_{\mathrm{e}}$ (their Definition 4.8), whose models are definably complete structures $(M,\leq,0,1,+,\cdot,\mathrm{e})$ where $\mathrm{e}:M\to M$ is differentiable on the interval $(-1,1)$ with $\mathrm{e}'=\mathrm{e}$ there, $\mathrm{e}(0)=1$, and $\mathrm{e}(x)=0$ outside $(-1,1)$ (a restricted exponential). Here $\mathbb{R}$ with the restricted exponential is a reduct of the structure of restricted analytic functions with exponentiation of van den Dries–Macintyre–Marker (1994), whose complete theory $T_{an}$ is polynomially bounded (every unary definable function is ultimately bounded by a polynomial) and model complete. By results of Fornasiero–Servi (2010) and Hieronymi (2013) relayed as Theorem 4.9 of the paper, every model of either $T^{dc}_{\exp}$ or $T^{dc}_{\mathrm{e}}$ is o-minimal, so the two theories sit inside the classical tame setting while being axiomatized without any reference to the real numbers.

The unconditional main result of Berarducci and Gallinaro (Theorem 12.4) is the model completeness of $T^{dc}_{\mathrm{e}}$. Its proof replaces the archimedean toolkit behind Wilkie's model-completeness proof (tools such as the Weierstrass preparation theorem are unavailable in non-archimedean, merely definably complete structures) with an induction on Khovanskii points — solutions of systems of $\mathrm{e}$-polynomial equations with non-vanishing Jacobian — carried out in the residue field of a local ring of germs of polynomially bounded definable functions, where Ax's (1971) functional transcendence theorem for differential fields supplies the needed transcendence control. Under $\mathrm{SC}_{\mathbb{R}}$ the authors then obtain completeness of $T^{dc}_{\mathrm{e}}$ (Theorem 14.2), by showing that every sufficiently saturated model contains a substructure elementarily equivalent to the real restricted exponential field; the role of $\mathrm{SC}_{\mathbb{R}}$ is precisely to guarantee, through the residue identification of the finite elements modulo the infinitesimals, that every model has an archimedean submodel of the right form. Completeness of $T^{dc}_{\exp}$ (Theorem 14.4) follows from completeness of $T^{dc}_{\mathrm{e}}$ by a transfer theorem of Ressayre (1993, Theorem 4.10 of the paper), so $T^{dc}_{\exp}=T_{\exp}$ and, the axioms being recursive, $T_{\exp}$ is decidable — recovering Macintyre–Wilkie. Under WSC the same argument yields completeness of the expanded theories $T^{dc}_{\mathrm{e}}+T_{wsc}$ and $T^{dc}_{\exp}+T_{wsc}$ (Remark 14.5(1)). Unconditionally, however, three questions remain, and the authors state each of them explicitly: whether $T^{dc}_{\mathrm{e}}$ (or at least $T^{dc}_{\exp}$) is complete without transcendence hypotheses; whether $T^{dc}_{\mathrm{e}}$ is polynomially bounded — since $T_{an}$ is polynomially bounded and complete, so is its reduct $T_{\mathrm{e}}$, but this cannot be pulled back to the subtheory $T^{dc}_{\mathrm{e}}\subseteq T_{\mathrm{e}}$ without completeness, as the authors point out; and whether, for a convex subring $\mathcal{O}$ with maximal ideal $\mathfrak{m}$ of a model $M\models T^{dc}_{\mathrm{e}}$, the residue field $\mathcal{O}/\mathfrak{m}$ with the induced restricted exponential $\bar{\mathrm{e}}$ of Corollary 5.3 is itself a model of $T^{dc}_{\mathrm{e}}$ — the authors write that they do not know this, since a priori the residue field may fail to be definably complete (the archimedean case $\mathcal{O}=\mathrm{Fin}(M)$ is settled affirmatively by Proposition 5.4 via embedding into the real restricted exponential field, and Theorem 15.1 gives a related embedding result reminiscent of the T-convexity theory of van den Dries–Lewenberg (1995)). The three questions are logically linked: by the discussion above, completeness of $T^{dc}_{\mathrm{e}}$ implies completeness of $T^{dc}_{\exp}$, hence $T^{dc}_{\exp}=T_{\exp}$ and unconditional decidability of $T_{\exp}$, hence WSC; it also implies polynomial boundedness of $T^{dc}_{\mathrm{e}}$; and a negative answer (two elementarily inequivalent models of $T^{dc}_{\mathrm{e}}$ or of $T^{dc}_{\exp}$) refutes $\mathrm{SC}_{\mathbb{R}}$ outright. Miller's (1996) growth dichotomy for o-minimal expansions of ordered fields — polynomially bounded, or defining exponentiation on a final segment — is the natural frame for the polynomial-boundedness question, and the authors suggest Miller's dichotomy and the quasianalytic tools of Rolin–Servi-type results as possible approaches while calling the problem rather challenging.

Problem Statement

Determine, without any transcendence hypothesis, the model-theoretic status of the definably complete exponential theories of Berarducci–Gallinaro (arXiv:2603.08365). The common objective is to remove the Schanuel-type assumptions from their completeness theorem. The source leaves open the following three questions:

Q1 (core, completeness). Working in the language $\{\leq,0,1,+,\cdot,\mathrm{e}\}$ (respectively $\{\leq,0,1,+,\cdot,\exp\}$), let $T^{dc}_{\mathrm{e}}$ be the theory whose models are the definably complete structures $(M,\leq,0,1,+,\cdot,\mathrm{e})$ with $\mathrm{e}$ differentiable on $(-1,1)$, $\mathrm{e}'=\mathrm{e}$ on $(-1,1)$, $\mathrm{e}(0)=1$, and $\mathrm{e}(x)=0$ for $x\notin(-1,1)$ (Definition 4.8 of the source), and let $T^{dc}_{\exp}$ be the theory whose models are the definably complete expansions of an ordered field with $\exp'=\exp$ and $\exp(0)=1$ (Definition 4.5). Is $T^{dc}_{\mathrm{e}}$ complete, i.e. does it prove either $\varphi$ or $\neg\varphi$ for every first-order sentence $\varphi$ of its language? Completeness of $T^{dc}_{\mathrm{e}}$ transfers to $T^{dc}_{\exp}$ (Ressayre's theorem as used in Theorem 14.4 of the source), so Q1 contains the weaker and more classical target: can $T^{dc}_{\exp}$ be proved complete — equivalently, since $T^{dc}_{\exp}\subseteq T_{\exp}$ is recursively axiomatized and $\mathbb{R}_{\exp}\models T^{dc}_{\exp}$, can $T^{dc}_{\exp}=T_{\exp}$, and hence the decidability of the theory of the real exponential field, be proved without assuming $\mathrm{SC}_{\mathbb{R}}$, WSC, or any other unproved transcendence statement? A positive answer for $T^{dc}_{\exp}$ constitutes an unconditional proof of the decidability of $T_{\exp}$ and therefore of WSC; a negative answer — two models of $T^{dc}_{\mathrm{e}}$ (or of $T^{dc}_{\exp}$) that are not elementarily equivalent, i.e. that disagree about some first-order sentence — refutes $\mathrm{SC}_{\mathbb{R}}$ by the contrapositive of Theorems 14.2 and 14.4 of the source.

Q2 (polynomial boundedness). Is $T^{dc}_{\mathrm{e}}$ polynomially bounded, i.e. is it the case that in every model $M\models T^{dc}_{\mathrm{e}}$ and for every $M$-definable function $f:M\to M$ there is $n\in\mathbb{N}$ with $|f(x)|\leq x^{n}$ for all sufficiently large $x\in M$? Note that a positive answer to Q1 (completeness of $T^{dc}_{\mathrm{e}}$) implies a positive answer here, since then $T^{dc}_{\mathrm{e}}$ coincides with the complete theory of the real restricted exponential field, a reduct of the polynomially bounded theory $T_{an}$; the question is open unconditionally, and also remains meaningful independently of Q1.

Q3 (residue fields). Let $M\models T^{dc}_{\mathrm{e}}$, let $\mathcal{O}\subset M$ be a convex subring (an order-convex subring) with maximal ideal $\mathfrak{m}$, and let $(\mathcal{O}/\mathfrak{m},\bar{\mathrm{e}})$ be the induced restricted exponential field of Corollary 5.3 of the source. Is $(\mathcal{O}/\mathfrak{m},\bar{\mathrm{e}})$ a model of $T^{dc}_{\mathrm{e}}$ — that is, since a priori the residue field may fail to be definably complete, is the expansion $(\mathcal{O}/\mathfrak{m},\bar{\mathrm{e}})$ definably complete? The archimedean case $\mathcal{O}=\mathrm{Fin}(M)$, where $\mathrm{Fin}(M)$ is the ring of finite elements (elements bounded in absolute value by a natural number), is settled by Proposition 5.4 of the source; the question concerns general convex subrings.

References

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