Wilkie's Theorem

Let $\phi(x_1,\ldots,x_n)$ be an ${\mathcal L}_{\rm exp}$ formula, where ${\mathcal L}_{\rm exp}\equiv {\mathcal L}\cup\{e^x\}$ and ${\mathcal L}$ is the language of ordered rings ${\mathcal L}=\{+, -, \cdot, <, 0, 1\}$. Then there are $n\geq m$ and $f_1, \ldots, f_s\in\Bbb{Z}[x_1, \ldots, x_n, e^{x_1}, \ldots, e^{x_n}]$ such that $\phi(x_1,\ldots,x_n)$ is equivalent to

$$\exists x_{m+1}\cdots\exists x_n f_1(x_1, \ldots, x_n, e^{x_... ...n})=\ldots = f_s(x_1, \ldots, x_n, e^{x_1}, \ldots, e^{x_n})=0$$

(Wilkie 1996). In other words, every formula is equivalent to an existential formula and every definable set is the projection of an exponential variety (Marker 1996).

References

Marker, D. ``Model Theory and Exponentiation.'' Not. Amer. Math. Soc. 43, 753-759, 1996.

Wilkie, A. J. ``Model Completeness Results for Expansions of the Ordered Field of Real Numbers by Restricted Pfaffian Functions and the Exponential Function.'' J. Amer. Math. Soc. 9, 1051-1094, 1996.