AI 中文总结
该研究借助切片秩与划分秩，将支撑熵方法应用于多项式零规避，得到高阶Erdős–Ginzburg–Ziv常数的均匀界，并首次给出F₅ⁿ上对应第四初等对称多项式的非平凡指数界。
AI 中文摘要
我们借助切片秩和划分秩研究有限向量空间上的多项式零规避问题。首先，我们通过证明当系数支撑不存在具有均匀边缘的概率分布时，有限对偶证书会产生显式熵间隙，使支撑熵方法变得有效。对于特征为3的域上的二次初等对称多项式，系数支撑的三元结构给出了具有最优归一化边缘的证书，以及对应高阶Erdős–Ginzburg–Ziv常数的均匀解析界，无需为每个域单独优化。随后，我们利用划分秩处理两两不同变量的解，将等式轮廓编码为收缩局部张量，把全局问题简化为有限多个切片秩估计。在乘法环面上应用此简化，得到了字母大小以下指数基的受限字母零和界。据我们所知，结合逐坐标逆变换与支撑分层，首次得到了与第四初等对称多项式相关的F₅ⁿ上高阶Erdős–Ginzburg–Ziv问题的非平凡指数界。
英文摘要
We study polynomial zero-avoidance over finite vector spaces by means of slice rank and partition rank. We first make the support-entropy method effective by showing how a finite dual certificate yields an explicit entropy gap whenever the coefficient support admits no probability distribution with uniform marginals. For the quadratic elementary symmetric polynomial over fields of characteristic three, the ternary structure of the coefficient support gives a certificate with optimal normalized margin and a uniform analytic bound for the corresponding higher-degree Erdős--Ginzburg--Ziv constant, avoiding a separate optimization for each field. We then use partition rank to handle solutions in pairwise distinct variables. Equality profiles are encoded by contracted local tensors, reducing the global problem to finitely many slice-rank estimates. Applying this reduction on the multiplicative torus gives restricted-alphabet zero-sum bounds with exponential base below the alphabet size. Coordinatewise inversion and support stratification then yield, to the best of our knowledge, the first nontrivial exponential bound for the higher-degree Erdős--Ginzburg--Ziv problem over $\mathbb{F}_5^n$ associated with the fourth elementary symmetric polynomial.
