数域上$n$-泛性格的渐近性
Asymptotics of $n$-universal lattices over number fields
AI总结：
本研究针对全实数域上的$n$-泛性格，推导了其最小秩对数的显式渐近公式，并证明对$n≥3$，秩或准则集规模不超过给定常数$C$的全实域仅有有限个且均可有效计算。
摘要：
我们证明了全实数域整数环上$n$-泛性格的最小秩的对数的显式渐近公式。我们还表明，对于任意常数$C > 0$且$n \geq 3$，仅存在有限多个具有秩至多为$C$的$n$-泛性格的全实域，且所有这类域均可有效计算。类似地，对于任意$n \geq 3$，我们证明仅存在有限多个容许大小至多为$C$的$n$-泛性准则集的全实域，所有这类域同样可有效计算。
英文摘要：
We prove an explicit asymptotic formula for the logarithm of the minimal ranks of $n$-universal lattices over the ring of integers of totally real number fields. We also show that, for any constant $C > 0$ and $n \geq 3$, there are only finitely many totally real fields with an $n$-universal lattice of rank at most $C$, with all such fields being effectively computable. Similarly, for any $n \geq 3$, we show that there are only finitely many totally real fields admitting an $n$-universal criterion set of size at most $C$, with all such fields likewise being effectively computable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
