Some bounds arising from a polynomial ideal associated to any $t$-design
Some bounds arising from a polynomial ideal associated to any $t$-design
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DOI：
10.13069/jacodesmath.729446
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发表时间：
2020-05
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通讯作者：
W. Martin;Douglas R Stinson
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作者：
W. Martin;Douglas R Stinson
我们考虑有序对 $(X,\mathcal{B})$ 在哪里 $X$ 是有限大小的集合吗 $v$ 和 $\mathcal{B}$ 是一些 $k$的元素子集 $X$ 这样每一个 $t$的-元素子集 $X$ 恰好包含在 $\lambda$ “积木” $B\in \mathcal{B}$ 对于一些固定的 $\lambda$． 我们表示每个块 $B$ 一个0 - 1向量 $\bc_B$ 长度 $v$ 探索理想 $\mathcal{I}(\mathcal{B})$ 的多项式 $v$ 具有复系数的变量在集合上消失 $\{ \bc_B \mid B \in \mathcal{B}\}$． 在建立了基本理论之后，我们研究了与这一理想相关的两个参数： $\gamma_1(\mathcal{B})$ 理想情况下非平凡多项式的最小次是多少 $\mathcal{I}(\mathcal{B})$ 和 $\gamma_2(\mathcal{B})$ 是最小的整数 $s$ 这样 $\mathcal{I}(\mathcal{B})$ 是由一组次数最多的多项式生成的吗 $s$． 我们首先证明一般界 $t/2 < \gamma_1(\mathcal{B}) \le \gamma_2(\mathcal{B}) \le k$． 检查重要的例子族，我们发现，对于对称2-设计和斯坦纳系统，我们有 $\gamma_2(\mathcal{B}) \le t$． 但是我们期望 $\gamma_2(\mathcal{B})$ 更接近 $k$ 对于非结构化设计，我们通过构造无限多个三重系统来表明这一点 $\gamma_2(\mathcal{B})=k$．
We consider ordered pairs $(X,\mathcal{B})$ where $X$ is a finite set of size $v$ and $\mathcal{B}$ is some collection of $k$-element subsets of $X$ such that every $t$-element subset of $X$ is contained in exactly $\lambda$ ``blocks'' $B\in \mathcal{B}$ for some fixed $\lambda$. We represent each block $B$ by a zero-one vector $\bc_B$ of length $v$ and explore the ideal $\mathcal{I}(\mathcal{B})$ of polynomials in $v$ variables with complex coefficients which vanish on the set $\{ \bc_B \mid B \in \mathcal{B}\}$. After setting up the basic theory, we investigate two parameters related to this ideal: $\gamma_1(\mathcal{B})$ is the smallest degree of a non-trivial polynomial in the ideal $\mathcal{I}(\mathcal{B})$ and $\gamma_2(\mathcal{B})$ is the smallest integer $s$ such that $\mathcal{I}(\mathcal{B})$ is generated by a set of polynomials of degree at most $s$. We first prove the general bounds $t/2 < \gamma_1(\mathcal{B}) \le \gamma_2(\mathcal{B}) \le k$. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have $\gamma_2(\mathcal{B}) \le t$. But we expect $\gamma_2(\mathcal{B})$ to be closer to $k$ for less structured designs and we indicate this by constructing infinitely many triple systems satisfying $\gamma_2(\mathcal{B})=k$.