发表机构
University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了布尔立方体上满足高阶消失条件（非零顶点至少k阶、原点恰好l阶）的多项式最小次数，给出了显式基并精确刻画了各特征下的特征下降。
AI 中文摘要
设 $F$ 为一个域，设 $0\le \ell\le k-2$，并假设 $n\ge k-1$。我们确定了 $F[x_1,\ldots,x_n]$ 中在布尔立方体的每个非零顶点处至少以阶 $k$ 消失、且在原点处以恰好阶 $\ell$ 消失的多项式的最小次数。答案为 \\[ n+2k-2-\rho_F(k-\ell), \\] 其中 $\rho_F(s)$ 是使对应的卡特兰数在 $F$ 中非零的正整数之和为 $s-1$ 的最小正整数个数。证明给出了约化消失空间的一个显式基。在这个基中，最高次映射是对角的，对角线上是卡特兰数；使用块坐标的细化表明该基与多项式次数相容。在奇特征下，次数为 $n+2k-3$ 或 $n+2k-4$，分别取决于 $C_{k-\ell-2}$ 是否非零。在特征 $2$ 下，次数为 \\[ n+2k-2-s_2(k-\ell-1), \\] 其中 $s_2$ 表示二进制数字和。因此，第一次特征下降以及所有后续下降都被精确确定。
英文摘要
Let $F$ be a field, let $0\le \ell\le k-2$, and suppose that $n\ge k-1$. We determine the minimum degree of a polynomial in $F[x_1,\ldots,x_n]$ that vanishes to order at least $k$ at every nonzero vertex of the Boolean cube and to order exactly $\ell$ at the origin. The answer is \[ n+2k-2-ρ_F(k-\ell), \] where $ρ_F(s)$ is the least number of positive integers summing to $s-1$ whose corresponding Catalan numbers are nonzero in $F$. The proof gives an explicit basis of the reduced vanishing space. In this basis the top-degree map is diagonal, with Catalan numbers on the diagonal; a refinement using block coordinates shows that the basis is compatible with polynomial degree. In odd characteristic the degree is either $n+2k-3$ or $n+2k-4$, according as $C_{k-\ell-2}$ is nonzero or zero. In characteristic $2$ it is \[ n+2k-2-s_2(k-\ell-1), \] where $s_2$ denotes binary digit sum. Thus the first characteristic drop, and all later drops, are determined exactly.
Comments15 pages
