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The rationality of the Poincaré series associated to the p-adic points on a variety.

J. Denef (1984)

Inventiones mathematicae

The real field with the rational points of an elliptic curve

Ayhan Günaydın, Philipp Hieronymi (2011)

Fundamenta Mathematicae

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.

Two theorems on regularly r-closed fields.

Yu.L. Ershov (1984)

Journal für die reine und angewandte Mathematik

Uniform p-adic cell decomposition and local zeta functions.

Johan Pas (1989)

Journal für die reine und angewandte Mathematik

Uniformization of rigid subanalytic sets

Hans Schoutens (1994)

Compositio Mathematica

Zero-one laws for graphs with edge probabilities decaying with distance. Part II

Saharon Shelah (2005)

Fundamenta Mathematicae

Let Gₙ be the random graph on [n] = 1,...,n with the probability of i,j being an edge decaying as a power of the distance, specifically the probability being $p_{| i - j |} = 1 / {| i - j |}^{α}$ , where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a “finite compactness”. Lastly, as a consequence, we prove that the zero-one law (for first order logic)...

Zero-one laws for graphs with edge probabilities decaying with distance. Part I

Saharon Shelah (2002)

Fundamenta Mathematicae

Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability $p_{| i - j |} = 1 / {| i - j |}^{α}$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..