Paper detail

Modules with many non-associates and norm form equations with many families of solutions

For every number field $\mathbb{K}$, with $[\mathbb{K}:\mathbb{Q}] \geq 3$, we show that the number of non-associates of the same norm in a full module in $\mathbb{K}$ does not depend only on $\mathbb{K}$, but can also depend on the module itself. As a corollary, the same can be true for the number of families of solutions of degenerate norm form equations. So the uniform bound obtained by Schmidt for the number of solutions in the non-degenerate case does not always hold in the degenerate case. For three-variable norm forms not arising from full modules, we do obtain a Schmidt-type bound for the number of families of solutions that, together with the above result, completes this aspect of the study of three-variable norm forms.