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Existence of irreducible principal series for α = 3ω + 2

Determine whether there exists an irreducible principal series b ∈ K((ℝ^{≤0})) with sup(supp(b)) = 0 and ot(b) = ω^{3ω+2}; equivalently, decide whether P_{3ω+2} contains any irreducible element.

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Background

The authors establish irreducibility for many principal series whose order types correspond to sums of up to three natural powers of ω, under various conditions on the Cantor normal form. They explicitly achieve the case α = 3ω + 1 but not α = 3ω + 2.

They point out that α = 3ω + 2 is currently the smallest ordinal α for which the existence of any irreducible element in P_α remains unknown, highlighting a concrete gap in the landscape of known irreducible principal series.

References

As a result, we shall prove irreducibility for many series in $P_\alpha$ for $\alpha = 3\omega + 1$ but not for $\alpha = 3\omega + 2$. In fact, $3\omega + 2$ is now the smallest $\alpha$ for which we do not know whether there exists any irreducible element in $P_{\alpha}$.

— Irreducibility in generalized power series (2405.13815 - Fornasiero et al., 22 May 2024) in Section 4, Subsection “The case ω^{α1} + ω^{α2} + ω^{α3}” (opening discussion)