We show that the universal theory of ${\rm SC_{def}}(K,d)$, in a natural expansion by definition of the lattice language, is the same for every such field $K$. We give a finite axiomatization of it and prove that it is locally finite and admits a model-completion, which turns to be decidable as well as all its completions. We expect $L({\mathbb Q}_p^d)$ to be a model of (a little variant of) this model-completion. This leads us to a new conjecture in $p$-adic semi-algebraic geometry which, combined with the results of this paper, would give decidability (via a natural recursive axiomatization) and elimination of quantifiers for the complete theory of $L({\mathbb R}_p^d)$, uniformly in $p$.

Cited literature [3 references]

https://hal.archives-ouvertes.fr/hal-00083404

Contributor : Luck Darnière <>

Submitted on : Friday, June 30, 2006 - 1:32:13 PM

Last modification on : Monday, March 9, 2020 - 6:15:51 PM

Long-term archiving on: : Monday, April 5, 2010 - 11:39:25 PM

Contributor : Luck Darnière <>

Submitted on : Friday, June 30, 2006 - 1:32:13 PM

Last modification on : Monday, March 9, 2020 - 6:15:51 PM

Long-term archiving on: : Monday, April 5, 2010 - 11:39:25 PM