Algebraic Dimension over Frobenius Fields

We prove that each perfect Frobenius field is algebraically bounded and hence has a dimension function in the sense of v.d. Dries on the collection of all definable sets. Given a definable set S over Q (resp. Fp) we can effectively determine for each k ∈ {−∞, 0, 1, . . .} whether there exists a perfect Frobenius fieldM of characteristic 0 (resp., of characteristic p) such that the dimension of S(M) is k. Our method of proof and decision procedure is based on Galois Stratification. Forum 6 (1994), 43–63 * This work was partially supported by a grant from the G.I.F., the German–Israeli Foundation for Scientific Research and Development, Tel Aviv University. It was partially done while the author visited the Institute for Experimental Mathematics in Essen.