Complexity and (un)decidability of fragments of $\langle \omega^{\omega^\lambda}; \times \rangle$

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