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Wilkie's Theorem

Let phi(x_1,...,x_m) be an L_(exp) formula, where L_(exp)=L union {e^x} and L is the language of ordered rings L={+,-,·,<,0,1}. Then there exist n>=m and f_1,...,f_s in Z[x_1,...,x_n,e^(x_1),...,e^(x_n)] such that phi(x_1,...,x_n) is equivalent to

exists x_(m+1)... exists x_nf_1(x_1,...,x_n,e^(x_1),...,e^(x_n))=... =f_s(x_1,...,x_n,e^(x_1),...,e^(x_n))=0

(Marker 1996, 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).

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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.

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Wilkie's Theorem

Cite this as:

Weisstein, Eric W. "Wilkie's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WilkiesTheorem.html

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