K, D is an (L, ∆)-precel l in K k, (cid:3)i denotes ‘<’ or ‘no condition’, and the ai (x) are functions from ∆k .
Cell decomposition and definable functions for weak p-adic structures
N, the set {x ∈ K k | ord a1 ≡ l mod n} can be partitioned as a ﬁnite union of precel ls ⊆ K k .
Cell decomposition and definable functions for weak p-adic structures
This completes the proof, since D is a precell, (3) is a precell condition and by our assumption, the set {x ∈ K k | ord a1 (x)λ−1 ≡ ζ mod n} can be partitioned as a ﬁnite union of precells.
Cell decomposition and definable functions for weak p-adic structures
L-deﬁnable functions and D is an (L, ∆)-precell.
Cell decomposition and definable functions for weak p-adic structures
The fact that quantiﬁer-free deﬁnable sets can be partitioned as a ﬁnite union of cells, also implies that for all ai ∈ ∆⊞R, K, the relation ord ai ≡ l mod n can be deﬁned using precell relations.
Cell decomposition and definable functions for weak p-adic structures
Any L∗ -precel l can be partitioned into a ﬁnite union of L∗ -cel ls.
Cell Decomposition for semibounded p-adic sets
However, up to a ﬁnite partition into precells, it is possible to write L∗ -polynomials in a more manageable way.
Cell Decomposition for semibounded p-adic sets
Therefore the condition ord gnk < l is equivalent to ord (vt + c(x)) < l, which is a precell condition.
Cell Decomposition for semibounded p-adic sets
How does this help us to express that ord gnk = 0? By the reasoning above, we can express > l using precell conditions.
Cell Decomposition for semibounded p-adic sets
To conclude the proof, partition K r+1 in precells A such that conditions of type (2) hold on each precell.
Cell Decomposition for semibounded p-adic sets
For precells on which f (x, t) is unbounded and has small enough order, this is clear.
Cell Decomposition for semibounded p-adic sets
On other precells, f (x, t) is bounded, so that we can prove our claim in exactly the same way as in case 1 of Section 2.5 of (again being careful about deﬁnability for multiplication and division).
Cell Decomposition for semibounded p-adic sets
Note that we can restrict our attention to those precells on which ord t > 0.
Cell Decomposition for semibounded p-adic sets
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