Polynomial bounds for solutions of a parametric cubic Thue equation

Prove that for every fixed constant C>0, or in particular for some fixed polynomial exponent C, all integer solutions (x,y) of x^3-ty^3=1 satisfy max{|x|,|y|}≤t^C for sufficiently large positive integers t.

Background

The paper contrasts its polynomial bounds for certain parametric norm-form equations with the unresolved equation x3-ty3=1, for which the authors state that no suitable functional units are available and standard linear-forms-in-logarithms estimates generally yield only exponential dependence on t. They further note that a well-known conjecture would imply the desired type of bound with any exponent C>1.

References

It may be worth remembering that no-one knows if for example $x3-ty3=1$ entails $\max{|x|,|y|} \leq tC$, even for $t$ large (by well-known results - see for example Bennett - there can be at most one solution with $y \neq 0$). Of course we have no suitable functional units, and linear forms in logarithms usually lead only to exponential polynomial dependence. On the third hand a well-known conjecture (almost equally well-known to be still open) implies that this holds for any $C > 1$.

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