Unconditional decidability of equality of exponential polynomials in $\pi$ (over $\mathbb{R}$) and in $\pi$ and $i$ (over $\mathbb{C}$)

D'Aquino and Terzo (2007) proved, modulo Schanuel's Conjecture, that there are algorithms deciding whether two exponential polynomials in $\pi$ are equal in $\mathbb{R}$ and whether two exponential polynomials in $\pi$ and $i$ coincide in $\mathbb{C}$ ; their normal-form argument collapses exactly because excluding nontrivial exponential-algebraic relations on $\pi$ (beyond $i^2=-1$ and $e^{i\pi}=-1$ over $\mathbb{C}$ ) is precisely what Schanuel's Conjecture supplies. The problem asks for the unconditional status of these decision problems: either an algorithm, with termination and correctness provable without unproved transcendence hypotheses, that decides equality of such closed exponential terms, or a proof that no such algorithm exists. Every known decidability result in this circle — from Macintyre's 1991 conditional solution of the word problem for exponential constants to Macintyre–Wilkie's 1996 conditional decidability of the full theory of the real exponential field, reproved conditionally by Berarducci–Gallinaro in 2026 — remains dependent on Schanuel-type assumptions, and even the subcase of deciding identities among pure exponential constants ( $e$ , $e^e$ , $e^{e^2}$ , …) has been open since 1991. The unconditional status is open.

Background

An exponential ring (E-ring) is a commutative ring $R$ with unit equipped with an exponential map $E:R\to R$ that is a homomorphism from the additive group $(R,+)$ to the multiplicative monoid $(R,\cdot)$ , so $E(u+v)=E(u)E(v)$ and $E(0)=1$ . The free E-ring $[x_1,\dots,x_m]_E$ on generators $x_1,\dots,x_m$ consists of all closed terms built from the integers and the generators using addition, additive inverses, multiplication, and $E$ ; Macintyre (1991) showed it has a primitive-recursive normal form, hence a primitive-recursive word problem, so equality of terms in the free E-ring is unconditionally decidable. An exponential polynomial in $\pi$ is such a closed term with the generator evaluated at $x=\pi$ , i.e., an element of the E-subring $\langle\pi\rangle_E$ of $\mathbb{R}$ generated by $\pi$ — for example $\pi^2 e^{e^{\pi}} + 3e^{2\pi}$ ; likewise exponential polynomials in $\pi$ and $i$ are elements of $\langle\pi,i\rangle_E\subseteq\mathbb{C}$ . Deciding equality of two such terms in $\mathbb{R}$ (or $\mathbb{C}$ ) is therefore a word problem for the kernel of the evaluation homomorphism from the free E-ring into $\mathbb{R}$ (or $\mathbb{C}$ ), and the difficulty is entirely in describing that kernel. Schanuel's Conjecture (SC), formulated around 1960, asserts that if $\lambda_1,\dots,\lambda_n\in\mathbb{C}$ are linearly independent over $\mathbb{Q}$ , then the field $\mathbb{Q}(\lambda_1,\dots,\lambda_n,e^{\lambda_1},\dots,e^{\lambda_n})$ has transcendence degree at least $n$ over $\mathbb{Q}$ ; it generalizes the Lindemann–Weierstrass theorem (exponentials of pairwise distinct algebraic numbers are linearly independent over the algebraic numbers) and is the standard, unproved input for all algebraic-independence information about exponential terms. Modulo SC, Macintyre (1991) proved that the exponential constants (the E-subring $\langle 1\rangle_E$ of $\mathbb{R}$ , containing $e$ , $e^e$ , $e^{e^2}$ , …) satisfy no hidden exponential-algebraic relations, so their equalities are decidable; D'Aquino and Terzo (2007) extended this to $\pi$ , showing modulo SC that $\pi$ is exponentially algebraically independent over $\mathbb{Z}$ , i.e. that no nonzero $p\in[x]_E$ vanishes at $\pi$ (trivial evaluation kernel over $\mathbb{R}$ ) and that over $\mathbb{C}$ the only relations between $i$ and $\pi$ are the two evident ones, $i^2=-1$ and $e^{i\pi}=-1$ ; in both cases comparison of free normal forms then decides equality. These results sit inside a well-known conditional landscape: Macintyre and Wilkie (1996) proved that SC implies decidability of the complete first-order theory of the real exponential field $(\mathbb{R};+,\cdot,\exp)$ — reproved, still modulo SC, by Berarducci and Gallinaro (2026) via definably complete exponential fields (ordered exponential fields in which every definable subset that is bounded above has a least upper bound) — while the unconditional decidability of that theory is a major open problem; on the computer-algebra side, Richardson (1968) proved undecidability of zero-testing for the richer class of expressions in a real variable involving $\sin$ — a proof that itself leans on Schanuel-type assumptions about the constants occurring in it — and Richardson (2007) showed that SC yields a decision procedure for equality among closed-form constants, while stating explicitly that without the conjecture "we do not yet know whether or not this is possible". The D'Aquino–Terzo question is the sharply delimited, purely exponential corner of this landscape: is the Schanuel assumption genuinely necessary there, or does some unconditional algorithm decide these particular closed-term identities?

Problem Statement

Let $[x]_E$ be the free exponential ring on one generator: the set of closed terms built from the integers and $x$ using addition, additive inverses, multiplication, and a unary function symbol $E$ satisfying $E(u+v)=E(u)E(v)$ and $E(0)=1$ ; by Macintyre (1991) it has a primitive-recursive normal form, hence a primitive-recursive word problem. Two decision problems are at issue. (a) The real problem: decide, given two terms $p,q\in[x]_E$ , whether $p(\pi)=q(\pi)$ holds in $\mathbb{R}$ , where $p(\pi)$ means evaluation of the generator at $x=\pi$ (so the values are elements of the E-subring $\langle\pi\rangle_E$ of $\mathbb{R}$ ). (b) The complex problem: the same question for closed terms $p,q$ in two generators evaluated at $x=\pi$ and $y=i$ , i.e. elements of $\langle\pi,i\rangle_E\subseteq\mathbb{C}$ ; under Schanuel's Conjecture the corresponding conditional normal form quotients exactly by $i^2+1=0$ and $e^{i\pi}+1=0$ . In both parts the algorithm's termination on every input and the correctness of every answer must be provable without unproved transcendence hypotheses — in particular without Schanuel's Conjecture. An affirmative answer consists of an explicit algorithm together with an unconditional proof of termination and correctness for all inputs (it need not prove that $\pi$ is exponentially algebraically independent; any unconditional decision procedure suffices; an algorithm for part (b) settles part (a) as well, since $\langle\pi\rangle_E$ embeds into $\langle\pi,i\rangle_E$ and equality of real values is reflected in $\mathbb{C}$ ; an algorithm for part (a) alone leaves part (b) open); a negative answer consists of a proof that no such algorithm exists; a proof of undecidability of part (a) settles part (b) as well, since (a) is a subcase of (b), while a proof of undecidability of part (b) alone leaves part (a) open. The problem asks for the existence of a uniform decision procedure, not the resolution of any individual instance; part (a) contains as a subcase the still-open unconditional word problem for the exponential constants — deciding whether two closed terms built from the integers alone, such as $e^{e^{2}}-e^{5}$ and $e^{2}+e^{-5}$ , are equal.

References

Paola D'Aquino and Giuseppina Terzo, 'A note on the decidability of exponential terms', Mathematical Logic Quarterly 53(3), 306–310 (2007). DOI: 10.1002/malq.200610047. https://doi.org/10.1002/malq.200610047
Angus Macintyre, 'Schanuel's conjecture and free exponential rings', Annals of Pure and Applied Logic 51(3), 241–246 (1991). DOI: 10.1016/0168-0072(91)90017-G. https://doi.org/10.1016/0168-0072(91)90017-G
Angus Macintyre and A. J. Wilkie, 'On the decidability of the real exponential field', in Piergiorgio Odifreddi (ed.), Kreiseliana: About and Around Georg Kreisel, A K Peters, Wellesley, MA, 441–467 (1996). ISBN 1-56881-061-X, Zbl 0896.03012. https://zbmath.org/1169378
Giuseppina Terzo, 'Consequences of Schanuel's Conjecture in exponential algebra', Tesi di dottorato, Università degli Studi di Napoli Federico II (2007). DOI: 10.6092/UNINA/FEDOA/2845. https://www.fedoa.unina.it/2845/
Paola D'Aquino, Angus Macintyre, and Giuseppina Terzo, 'Schanuel Nullstellensatz for Zilber fields', Fundamenta Mathematicae 207(2), 123–143 (2010). DOI: 10.4064/fm207-2-2. https://doi.org/10.4064/fm207-2-2
Paola D'Aquino, Angus Macintyre, and Giuseppina Terzo, 'Comparing ℂ and Zilber's exponential fields: zero sets of exponential polynomials', Journal of the Institute of Mathematics of Jussieu 15(1), 71–84 (2016). DOI: 10.1017/S1474748014000231, arXiv:1310.6891. https://arxiv.org/abs/1310.6891
Paola D'Aquino, Antongiulio Fornasiero, and Giuseppina Terzo, 'Generic solutions of equations with iterated exponentials', Transactions of the American Mathematical Society 370(2), 1393–1407 (2018). DOI: 10.1090/tran/7206, arXiv:1602.02016. https://arxiv.org/abs/1602.02016
Paola D'Aquino, Antongiulio Fornasiero, and Giuseppina Terzo, 'A weak version of the strong exponential closure', Israel Journal of Mathematics 242(2), 697–705 (2021). DOI: 10.1007/s11856-021-2141-1, arXiv:1909.10241. https://arxiv.org/abs/1909.10241
Antongiulio Fornasiero and Giuseppina Terzo, 'Note on radical and prime E-ideals', preprint arXiv:2309.02890 (2023). https://arxiv.org/abs/2309.02890
Paola D'Aquino, Antongiulio Fornasiero, and Giuseppina Terzo, 'E-ideals in exponential polynomial rings', Communications in Algebra 53(10), 4425–4448 (2025). DOI: 10.1080/00927872.2025.2486395, arXiv:2212.14347. https://arxiv.org/abs/2212.14347
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Daniel Richardson, 'Some undecidable problems involving elementary functions of a real variable', Journal of Symbolic Logic 33(4), 514–520 (1968). DOI: 10.2307/2271358. https://doi.org/10.2307/2271358
Daniel Richardson, 'Zero Tests for Constants in Simple Scientific Computation', Mathematics in Computer Science 1(1), 21–37 (2007). DOI: 10.1007/s11786-007-0002-x. https://doi.org/10.1007/s11786-007-0002-x
Alessandro Berarducci and Francesco Gallinaro, 'On the elementary theory of the real exponential field', preprint arXiv:2603.08365 [math.LO] (2026). https://arxiv.org/abs/2603.08365

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Unconditional decidability of equality of exponential polynomials in $\pi$ (over $\mathbb{R}$ ) and in $\pi$ and $i$ (over $\mathbb{C}$ )

Background

Problem Statement

References