Characterization of norm forms via their values at integer points

TL;DR

This paper characterizes certain algebraic forms based on their integer values, revealing cases where classical conjectures do not hold, using a homogeneous dynamical approach.

Contribution

It provides a complete description of forms with discrete integer values and non-trivial zero representations, highlighting exceptions to classical conjectures.

Findings

Forms with discrete integer values are characterized.

Certain non-purely real forms violate classical conjectures.

Homogeneous dynamical methods are effective for this classification.

Abstract

Using homogeneous dynamical approach, we obtain a complete description of the forms with discrete set of values at the integer points and not representing zero non-trivially over the rational numbers. As a consequence, we obtain a general class of non-purely real forms for which the natural generalization of Cassels and Swinnerton-Dyer conjecture fails.