Complexity and (Un)decidability of Fragments of 〈 ω ω λ;× 〉

We specify the frontier of decidability for fragments of the first-order theory of ordinal multiplication. We give a NEXPTIME lower bound for the complexity of the existential fragment of 〈 ω ω λ ; × , ω , ω +1, ω 2 + 1 〉 for every ordinal λ. Moreover, we prove (by reduction from Hilbert Tenth Problem) that the ∃*∀⁶-fragment of 〈 ω ω λ ; × 〉 is undecidable for every ordinal λ.