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

Abstract: 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 $\exists<sup>*\forall<sup>{6}$-fragment of $$\langle \omega^{{\omega^\lambda};} \times \rangle$$ is undecidable for every ordinal $\lambda$.