Positive-definite ternary quadratic forms with the same representations over ℤ

Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over [Formula: see text], they are equivalent over [Formula: see text] or constant multiples of regular forms, or is included in either of two families parameterized by [Formula: see text]. Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. First, the result of an exhaustive search for such pairs of integral quadratic forms is presented in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000 by computation. However, a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field is [Formula: see text] or [Formula: see text]. Second, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over [Formula: see text], their constant multiples are equivalent over [Formula: see text]. This was motivated by the question why the other families were not detected in the search. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms.

Orderable groups, elementary theory, and the Kaplansky conjecture

We show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We then consider the relationship with the Kaplansky group rings conjecture and show that {{\mathcal{K}}} , the class of groups which satisfy the conjecture, is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in {{\mathcal{K}}} or more generally two torsion-free groups are universally equivalent.