Verification of the unit-index condition for the septic polynomial X^7+27T^6X-1

Determine whether the unit-index equality [\xi_t]^*=([T,\xi]^*)_t holds for sufficiently large positive integers t when P(T,X)=X^7+27T^6X-1.

Background

The main theorem removes the need to verify the unit-index condition in the examples to which it applies, but the paper explicitly records that this condition is unknown for the septic polynomial P(T,X)=X7+27T6X-1. The unresolved condition concerns whether every unit in the specialized number field is generated, up to the relevant specialization, by the functional units.

References

Here Assumption \ref{ass:Thomas-H} is not too difficult to check. But now we do not know~eq:index.

— Pencils of norm form equations and a conjecture of Thomas, II  (2609.09995 - Amoroso et al., 9 Sep 2026) in Section examples, discussion of equation (3)