AI 中文总结
本文引入行列式容许集框架，将Bhargava阶乘构造推广为行列式线形式，通过多类阶乘实例发展理论，并提出p≡1 mod3素数的相关行列式构造，剩余有限位行列式的构造是待解决的开放问题。
AI 中文摘要
Bhargava的阶乘构造将戴德金整环的子集S与由局部p-序决定的阶乘理想序列关联起来。我们引入行列式容许性，这是一个阶乘演算框架，其局部序数据容许典范进位、轨道或更新模型，以及相容的相对行列式。对于此类集合，伽马函数的自然类似物通常不是标量值函数，而是一个完备行列式线。在清除有理谱重数后，其整数纤维恢复Bhargava阶乘理想的有限张量幂。在数域上，该线带有阿基米德度量；在全局函数域上，它在无穷远点处完备。在这一表述中，斯特林型公式源于带度量行列式的大参数渐近，反射公式源于对应行列式复形的对偶性。我们通过经典阶乘、二次多项式像、几何阶乘与q阶乘、Polya–Ostrowski阶乘理想、Carlitz–Goss阶乘，以及Bhargava–Barnes提升（其二级行列式恢复n-最优集的最优范德蒙德能量）发展该理论。立方数集是第一个真正的更新论例子。对于素数p ≡ 1 mod 3，我们构造了一个精确三态进位行列式，而通过二次特征χ_{-3}投影得到阿基米德半行列式Γ_{C,∞}(z) = √(Γ(z)Γ(3z−2))和一个自然平方根行列式线。剩余有限位贡献由素数p ≡ 2 mod 3处的非线性更新系统及p=3处的分歧系统支配；构造它们的典范全局相对行列式是主要的开放分析问题。
英文摘要
Bhargava's factorial construction attaches to a subset $S$ of a Dedekind domain a sequence of factorial ideals determined by local $p$-orderings. We introduce determinant-admissibility, a framework for factorial calculi whose local ordering data admit canonical carry, orbit, or renewal models together with compatible relative determinants. For such sets, the natural analogue of the Gamma function is generally not a scalar-valued function but a completed determinant line. After clearing rational spectral multiplicities, its integer fibers recover a finite tensor power of the Bhargava factorial ideals. Over number fields this line carries Archimedean metrics, while over global function fields it is completed at the distinguished places at infinity. In this formulation, Stirling-type formulas arise from large-parameter asymptotics of metrized determinants, and reflection formulas arise from dualities of the corresponding determinant complexes. We develop the theory through classical factorials, quadratic polynomial images, geometric and $q$-factorials, Polya--Ostrowski factorial ideals, Carlitz--Goss factorials, and a Bhargava--Barnes lift whose second-level determinant recovers the optimal Vandermonde energy of $n$-optimal sets. The set of cubes provides the first genuinely renewal-theoretic example. For primes $p \equiv 1 \pmod{3}$, we construct an exact three-state carry determinant, while projection through the quadratic character $χ_{-3}$ yields the Archimedean half-determinant \[ Γ_{C,\infty}(z) = \sqrt{Γ(z)Γ(3z-2)} \] and a natural square-root determinant line. The remaining finite-place contribution is governed by nonlinear renewal systems at primes $p \equiv 2 \pmod{3}$ and by a ramified system at $p=3$; constructing their canonical global relative determinant is the principal open analytic problem.
