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We specify the frontier of decidability for fragments of the first-ordertheory of ordinal multiplication. We give a NEXPTIME lower bound for thecomplexity 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 $\lambda$.

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