On ternary positive-definite quadratic forms with the same representations over Z

TL;DR

This paper investigates Kaplansky's conjecture on positive-definite ternary quadratic forms with identical representations over integers, combining computational searches and theoretical proofs to understand their structure and limitations.

Contribution

It provides an exhaustive computational search for such forms and proves that pairs with identical rational representations are essentially multiples of equivalent forms.

Findings

Few non-regular forms share identical representations up to 3,000,000.

Strong limitations exist on the existence of such pairs.

Pairs with identical rational representations are multiples of equivalent forms.

Abstract

Kaplansky conjectured that if two positive-definite real ternary quadratic forms have perfectly identical representations over $\mathbb{Z}$, they are constant multiples of regular forms, or is included in either of two families parametrized by $\mathbb{R}^2$ (so called, hexagonal and rhombohedral families). Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. Firstly, 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 are confirmed to have the identical representations up to 3,000,000, although a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field is $\mathbb{Q}$ or $\mathbb{R}$. Secondly, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over $\mathbb{Q}$, their constant multiples are equivalent over $\mathbb{Q}$. 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.