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Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function

A. Wilkie

Published 1996 in Journal of the American Mathematical Society

ABSTRACT

Recall that a subset of R is called semi-algebraic if it can be represented as a (finite) boolean combination of sets of the form {~ α ∈ R : p(~ α) = 0}, {~ α ∈ R : q(~ α) > 0} where p(~x), q(~x) are n-variable polynomials with real coefficients. A map from R to R is called semi-algebraic if its graph, considered as a subset of R, is so. The geometry of such sets and maps (“semi-algebraic geometry”) is now a widely studied and flourishing subject that owes much to the foundational work in the 1930s of the logician Alfred Tarski. He proved ([11]) that the image of a semi-algebraic set under a semi-algebraic map is semi-algebraic. (A familiar simple instance: the image of {〈a, b, c, x〉 ∈ R : a 6= 0 and ax +bx+c = 0} under the projection map R×R→ R is {〈a, b, c〉 ∈ R : a 6= 0 and b−4ac ≥ 0}.) Tarski’s result implies that the class of semi-algebraic sets is closed under firstorder logical definability (where, as well as boolean operations, the quantifiers “∃x ∈ R . . . ” and “∀x ∈ R . . . ” are allowed) and for this reason it is known to logicians as “quantifier elimination for the ordered ring structure on R”. Immediate consequences are the facts that the closure, interior and boundary of a semialgebraic set are semi-algebraic. It is also the basis for many inductive arguments in semi-algebraic geometry where a desired property of a given semi-algebraic set is inferred from the same property of projections of the set into lower dimensions. For example, the fact (due to Hironaka) that any bounded semi-algebraic set can be triangulated is proved this way. In the 1960s the analytic geometer Lojasiewicz extended the above theory to the analytic context ([8]). The definition of a semi-analytic subset of R is the same as above except that for the basic sets the p(~x)’s and q(~x)’s are allowed to be analytic functions and we only insist that the boolean representations work locally around each point of R (allowing different representations around different points). It is also necessary to restrict the maps to be proper (with semi-analytic graph). With this restriction it is true that the image of a semi-analytic set, known as a sub-analytic set, is semi-analytic provided that the target space is either R or R. Counterexamples have been known since the beginning of this century for maps to R for m ≥ 3. (They are due to Osgood, see [8].) However, the situation was clarified in 1968 by Gabrielov ([5]) who showed that the class of sub-analytic sets

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