On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)
On the number of monogenizations of a quartic order (with an appendix by Shabnam Akhtari)
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关于四次阶的单射数（Shabnam Akhtari 的附录）
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发表时间：
2021
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通讯作者：
M. Bhargava
中科院分区：
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作者：
M. Bhargava
我们证明了在一个四次域中的一个序作为一个Z -代数有少于3000个本质不同的生成元（如果该序的判别式足够大，则少于200个）。这大大提高了以前最著名的界限2 72。类似地，我们证明了四次域中的一个阶同构于至多10类整数二元四次型的不变阶（如果判别式足够大，则至多7类）。这大大提高了以前最著名的界限2 80。
We show that an order in a quartic field has fewer than 3000 essentially different generators as a Z -algebra (and fewer than 200 if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of 2 72 . Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most 10 classes of integral binary quartic forms (and at most 7 if the discriminant is sufficiently large). This significantly improves the previously best known bound of 2 80 .