Model theory of the real exponential function

A branch of model theory studying the elementary theory of the ordered field of real numbers with the real exponential function (cf. Exponential function, real). It is motivated by Tarski's question [a7], p. 45, whether is decidable.

A. Wilkie showed in [a8] that is model complete. Combining this with Khovanskii's finiteness theorem [a5], it follows that this theory is -minimal. In fact, Wilkie first studies expansions (cf. Structure) of by a Pfaffian chain of functions (see also [a2]): Fix and an open set containing the closed unit box . A Pfaffian chain of functions on is a sequence of analytic functions (cf. Analytic function) for which there exist polynomials (for ; ) such that

for all . Wilkie shows that the expansion of by a Pfaffian chain of functions restricted to the closed unit box has a model-complete theory. In particular, the expansion of by the restricted exponential function has a model-complete theory. Wilkie then deduces the model completeness of from this last result. An alternative proof of the model completeness, and an axiomatization of over , was found by J.P. Ressayre in 1991 (see [a3] for a generalization of Ressayre's result).

In [a6], A. Macintyre and Wilkie show that is decidable provided that the real version of Schanuel's conjecture (cf. Algebraic independence) is true.

The theory does not admit elimination of quantifiers. In fact, an expansion of by a family of total real-analytic functions (see [a1]) admits elimination of quantifiers if and only if each function is semi-algebraic, i.e., has a semi-algebraic graph (cf. Semi-algebraic set). However, let denote the family of restricted real-analytic functions, i.e., functions , for all , which are given on by a power series converging on a neighbourhood of and are set equal to outside of . It is shown in [a3] that the expansion admits elimination of quantifiers. The authors also give a complete axiomatization of , and establish that it is -minimal. In [a4] they construct a model of this theory which is not Archimedean and use it to solve a problem raised by G.H. Hardy: they show that the compositional inverse of the function is not asymptotic at to a composition of semi-algebraic functions, and .

References

[a1]	L. van den Dries, "Remarks on Tarski's problem concerning $({\bf R},+,\cdot,\exp)$" G. Lolli (ed.) G. Longo (ed.) A. Marcja (ed.) , Logic Colloquium '82 , North-Holland (1984) pp. 97–121
[a2]	L. van den Dries, "Tarski's problem and Pfaffian functions" J.B. Paris (ed.) A.J. Wilkie (ed.) G.M. Wilmers (ed.) , Logic Colloquium '84 , North-Holland (1986) pp. 59–90
[a3]	L. van den Dries, A.J. Macintyre, D. Marker, "The elementary theory of restricted analytic fields with exponentiation" Ann. of Math. , 140 (1994) pp. 183–205
[a4]	L. van den Dries, A.J. Macintyre, D. Marker, "Logarithmic-exponential power series" J. London Math. Soc. 56 (1997) 417-434. Zbl 0924.12007
[a5]	A. Khovanskii, "On a class of systems of transcendental equations" Soviet Math. Dokl. , 22 (1980) pp. 762–765 (In Russian)
[a6]	A.J. Macintyre, A.J. Wilkie, "On the decidability of the real exponential field" P.G. Odifreddi (ed.) , Kreisel 70th Birthday Volume , CLSI (1995)
[a7]	A. Tarski, J.C.C. McKinsey, "A decision method for elementary algebra and geometry" , Univ. California Press (1951)
[a8]	A.J. Wilkie, "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 : 4 (1996)