On positive-definite ternary quadratic forms with the same representations over Z
On positive-definite ternary quadratic forms with the same representations over Z
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关于 Z 上具有相同表示的正定三元二次形式
DOI：
10.1142/s1793042120500785
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
R. Oishi-Tomiyasu
中科院分区：
文献类型：
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作者：
Daigo Takano; Teruya Minamoto;R. Oishi-Tomiyasu
Kaplansky猜想，如果两个正定的三元二次型在上有完全相同的表示，则它们是正则形式的等价超常数倍，或者包含在参数化族中的任一族中。我们的结果旨在澄清通过计算和理论方法对这种配对施加的限制。首先，为了提供Kaplansky猜想的一个具体版本，我们给出了这种积分二次型对的穷尽搜索的结果。所获得的列表包含少量非规则形式，通过计算确认它们具有高达3,000,000个相同的表示。然而，无论系数场是非对称的，这种对的存在仍然存在很大的限制。其次，我们证明了如果两对三元二次型在上具有相同的同时表示，则它们的常数倍数在上是等价的。这样做的动机是，为什么在搜查中没有发现其他家庭。在证明中，利用Bhargava对四次环及其预解环的参数化，讨论了三元二次型对。
Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over, they are equivalent overor constant multiples of regular forms, or is included in either of two families parameterized by. 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 isor. Second, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over, their constant multiples are equivalent over. 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.