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

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 $\langle \omega^{{\omega\lambda};} \times, \omega, \omega+1, \omega²⁺¹ \rangle$ for every ordinal $\lambda$. Moreover, we prove (by reduction from Hilbert Tenth Problem) that the $\exists^{*\forall{6}$-fragment} of $\langle \omega^{{\omega\lambda};} \times \rangle$ is undecidable for every ordinal $\lambda$.