用多重性覆盖超立方体的几乎所有层
Almost covering all the layers of hypercube with multiplicities
AI总结：
研究通过多项式覆盖超立方体不同层的多重性，证明了多项式次数下限并构造了覆盖方案，同时推翻了关于对称子集的猜想。
摘要：
给定超立方体Q^n={0,1}^n和k∈{0,…,n}，k层Q^n_k是坐标中恰好有k个1的点集。对于固定的t∈N和k∈{0,…,n}，设P∈R[x₁,…,xₙ]为多项式，其在Q^n\Q^n_k处有至少t重零点，在Q^n_k处有恰好t−1重零点。本文证明deg(P)≥max{k,n−k}+2t−2。匹配该下界，我们构造了超平面族H₁,…,Hₘ，其中m=max{k,n−k}+2t−2，使得Q^n_k中的每个点被覆盖恰好t−1次，其他点被覆盖至少t次。当k=0,t=1时，恢复了Alon和Füredi的结果。利用该超平面族推翻了Venkitesh关于对称子集的猜想。我们还研究了一个新的受限和集问题变体。
英文摘要：
Given a hypercube $\mathcal{Q}^{n} := \{0,1\}^{n}$ in $\mathbb{R}^{n}$ and $k \in \{0, \dots, n\}$, the $k$-th layer $\mathcal{Q}^{n}_{k}$ of $\mathcal{Q}^{n}$ denotes the set of all points in $\mathcal{Q}^{n}$ whose coordinates contain exactly $k$ many ones. For a fixed $t \in \mathbb{N}$ and $k \in \{0, \dots, n\}$, let $P \in \mathbb{R}\left[x_{1}, \dots, x_{n}\right]$ be a polynomial that has zeroes of multiplicity at least $t$ at all points of $\mathcal{Q}^{n} \setminus \mathcal{Q}^{n}_{k}$, and $P$ has zeros of multiplicity exactly $t-1$ at all points of $\mathcal{Q}^{n}_{k}$. In this short note, we show that $$deg(P) \geq \max\left\{ k, n-k\right\}+2t-2.$$Matching the above lower bound we give an explicit construction of a family of hyperplanes $H_{1}, \dots, H_{m}$ in $\mathbb{R}^{n}$, where $m = \max\left\{ k, n-k\right\}+2t-2$, such that every point of $\mathcal{Q}^{n}_{k}$ will be covered exactly $t-1$ times, and every other point of $\mathcal{Q}^{n}$ will be covered at least $t$ times. Note that putting $k = 0$ and $t=1$, we recover the much celebrated covering result of Alon and Füredi (European Journal of Combinatorics, 1993). Using the above family of hyperplanes we disprove a conjecture of Venkitesh (The Electronic Journal of Combinatorics, 2022) on exactly covering symmetric subsets of hypercube $\mathcal{Q}^{n}$ with hyperplanes. To prove the above results we have introduced a new measure of complexity of a subset of the hypercube called index complexity which we believe will be of independent interest. We also study a new interesting variant of the restricted sumset problem motivated by the ideas behind the proof of the above result.
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