七阶幻方的埃尔哈特级数
The Ehrhart series of magic squares of orders seven and eight
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中文总结 AI 辅助
研究七阶幻方的埃尔哈特级数，通过SimpCone分解等方法确定其为既约有理函数，利用多种技术计算相关多面体在有限域上的生成函数等，最终确定了级数的分母、分子等特性并证明相关结论。
中文摘要 AI 辅助
设\(\mathrm{IMS}_n(m)\)表示\(n\times n\)非负整数矩阵的数量，其行和、列和以及两条主对角线和都等于\(m\)。我们将埃尔哈特级数\(F_7(q)=\sum_{m\ge 0}\mathrm{IMS}_7(m)q^m\)确定为一个既约有理函数。分母次数为\(373\)且阶数至多为\(15\)的分圆因子；分子是一个次数为\(366\)且系数为非负整数的回文多项式。利用SimpCone分解，相关多面体表示为\(1.66\)亿个带符号单纯锥的和。LRQC评估器计算它们在有限域上的生成函数；一个典型锥只需要一两个商特征，每个特征的成本在截断度\(T\)上几乎是线性的。明确的公分母和埃尔哈特互反性将有理重构简化到\(T = 1256\)的前缀，而明确的计数界限提供了从素域到\(\mathbb{Z}\)确定性提升所需的系数界限。这个前缀在八个素域中为整个锥族独立计算。精确的中国剩余定理将验证的残数提升到\(\mathbb{Z}\)上的等式，有限前缀准则证明有理恒等式，精确的多项式最大公因数证明所显示的分母是既约的。
英文摘要
Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator--denominator degrees are respectively $(366,373)$ and $(540,548)$. Both numerators have positive integer coefficients, are palindromic and strictly unimodal. The proofs share one finite architecture: a signed SimpCone decomposition is evaluated by quotient characters over finite fields, a certified common denominator and Ehrhart reciprocity reduce the rational identity to finitely many coefficients, an explicit counting bound lifts modular congruences to integer equalities, and exact gcd computations prove reducedness. For order eight, a face-index pole certificate gives a degree-$598$ common denominator without enumerating the full face lattice, leaving $296$ independent coefficients in degrees $0$ through $295$. Once the candidate rational function is known, the first six production primes certify this finite prefix by the same bounded-coefficient argument used for order seven.
