Invariants of Semi-algebraic p-adic Sets

R. Cluckers

(Joint work with D. Haskell, Department of mathematics, College of the Holy Cross, Worcester, Ma, USA).

We consider p-adic semi-algebraic sets, that is, definable subsets of ℚ_{p}^{k} in the language of fields with n-th power predicates in which we have quantifier elimination [M]. We prove via an explicit construction that there is a definable bijection between ℤ_{p} and ℤ_{p} \ {0}. It follows that there is no non-trivial Euler characteristic on ℚ_{p}. This result can analogously be obtained for a large class of Henselian valued fields with appropriate language. More generally, we use the cell decomposition [D] to prove that there is a definable bijection between two definable sets if and only if they have the same dimension (as defined by [SD]). The proof works also for finite field extensions of ℚ_{p}.

[D] Denef, J.: p-adic semi-algebraic sets and cell decomposition, J. Reine Angew. Math. 369 (1986), 154-166.

[M] Macintyre, A.: On definable subsets of p-adic fields, J. Symb. Logic 41 (1976), 605-610.

[SD] Scowcroft, P.; van den Dries, L.: On the structure of semialgebraic sets over p-adic fields J. Symbolic Logic 53 (1988), no. 4, 1138-1164.

Address: Department of Mathematics, University of Leuven, Celestijnenlaan 200B, 3001 Leuven, BEL-GIUM

E-mail: raf.cluckers@wis.kuleuven.ac.be