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 $\langle \omega^{\omega^\lambda}; \times, \omega, \omega+1, \omega^2+1 \rangle$ 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 $\lambda$.

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