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.

中文翻译：

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在 ℤ 上具有相同表示的正定三元二次型

卡普兰斯基推测，如果两个正定三元二次型在 [公式：见正文] 上具有完全相同的表示，则它们在 [公式：见正文] 或正则形式的常数倍数上是等价的，或者包含在参数化的两个族中的任何一个中[公式：见正文]。我们的结果旨在阐明计算和理论方法对这样一对施加的限制。首先，为了提供卡普兰斯基猜想的具体版本，给出了对此类积分二次形式对的详尽搜索的结果。获得的列表包含少量非正则形式，通过计算确认它们具有相同的表示形式，最多可达 3,000,000。然而，仍然观察到对这种对的存在的强烈限制，不管系数字段是[公式：见正文]还是[公式：见正文]。其次，我们证明如果两对三元二次型在[公式：见正文]上具有相同的同时表示，则它们的常数倍数在[公式：见正文]上是等价的。这是出于为什么在搜索中没有发现其他家庭的问题。在证明中，使用 Bhargava 对四次环及其分解环的参数化来讨论三元二次型对。这是出于为什么在搜索中没有发现其他家庭的问题。在证明中，使用 Bhargava 对四次环及其分解环的参数化来讨论三元二次型对。这是出于为什么在搜索中没有发现其他家庭的问题。在证明中，使用 Bhargava 对四次环及其分解环的参数化来讨论三元二次型对。