An existential 0-definition of F_q[[t]] in F_q((t))

We show that the valuation ring F_q[[t]] in the local field F_q((t)) is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an existential-F_q-definable bounded neighbouhood of 0. Then we `tweak' this set by subtracting, taking roots, and applying Hensel's Lemma in order to find an existential-F_q-definable subset of F_q[[t]] which contains tF_q[[t]]. Finally, we use the fact that F_q is defined by the formula x^q-x=0 to extend the definition to the whole of F_q[[t]] and to rid the definition of parameters. Several extensions of the theorem are obtained, notably an existential 0-definition of the valuation ring of a non-trivial valuation with divisible value group.Comment: 9 page

NONDEFINABILITY RESULTS FOR ELLIPTIC AND MODULAR FUNCTIONS

Let Ω be a complex lattice which does not have complex multiplication and ℘ = ℘Ω the Weierstrass ℘-function associated to it. Let D ⊆ C be a disc and I ⊆ R be a bounded closed interval such that I ∩ Ω = ∅. Let f : D → C be a function definablein (R, ℘|I ). We show that if f is holomorphic on D then f is definable in R. The proofof this result is an adaptation of the proof of Bianconi for the Rexp case. We also givea characterization of lattices with complex multiplication in terms of definability and a nondefinability result for the modular j-function using similar methods.<br/

Local interdefinability of Weierstrass elliptic functions

We explain which Weierstrass elliptic functions are locally definable from other elliptic functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way