范德蒙德行列式的幂最终是非 SNP 的
Powers of the Vandermonde determinant are eventually non-SNP
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中文总结 AI 辅助
证明范德蒙德行列式每个固定正幂在足够多变量中是非 SNP 的猜想，对偶数幂$k\geq4$，通过戴森常数项恒等式等展示牛顿多胞体中系数为零的格点，奇数情况由交错性得出，二次情况已知，关键构造和策略源于语言模型提示。
中文摘要 AI 辅助
我们证明了莫尼卡尔、托坎和勇的一个猜想，即范德蒙德行列式的每个固定正幂在足够多变量中都是非 SNP 的，其中如果一个多项式的牛顿多胞体中有一个格点没有以非零系数出现，则该多项式是非 SNP 的。这意味着我们的结果证明，对于每个偶数幂$k\geq4$，在足够大的维度中总是存在这样一个缺失的格单项式。奇数情况由交错性得出，二次情况先前已知。对于每个偶数幂$k\geq4$，我们在$a_{\delta_k}^k$的牛顿多胞体中展示了一个系数为零的显式格点。该消失性由戴森常数项恒等式得出，使用有限变量杰克标量积和麦克唐纳特殊化公式证明。关键的偶数幂构造和证明策略来自于由大型语言模型 OpenAI Codex（GPT Sol 5.6 超高）的提示；完整记录出现在附录中。作者随后检查并整理了论证。随附的精益形式化可在这个 https 网址获得。
英文摘要
We prove a conjecture of Monical, Tokcan, and Yong that every fixed positive power of the Vandermonde determinant is non-SNP in all sufficiently many variables, where a polynomial is non-SNP if there is a lattice point in its Newton polytope that does not appear with nonzero coefficient. This means our result proves that for every even power $k\geq4$, there is always such a missing lattice monomial in large enough dimensions. The odd case follows from alternation, and the quadratic case was previously known. For every even power $k\geq4$, we exhibit an explicit lattice point in the Newton polytope of $a_{δ_k}^k$ whose coefficient vanishes. The vanishing is obtained from a Dyson constant-term identity, proved using the finite-variable Jack scalar product and Macdonald's specialization formula. The key even-power construction and proof strategy arose from prompting with OpenAI Codex (GPT Sol 5.6 Extra High), a large language model; the complete transcript appears in the appendix. The authors subsequently checked and organized the argument. The accompanying Lean formalization is available at https://github.com/steven-le-thien/vandermonde-snp.
