切片上布尔函数的配对构造
The paired construction for Boolean functions on the slice
AI总结：
本文研究Johnson概形切片上由两个对立基本函数求和得到的配对布尔函数，通过初等方法确定了其精确多项式次数，证明在n=2k且偶次数t∉{0,k}时配对函数是已知最小非零布尔函数，优于t-pencils。
摘要：
设$V$是大小为$n$的有限集。我们研究“切片”$\binom{V}{k}$上的实函数，这类函数也被称为Johnson概形中的函数。对于$I \subseteq J \subseteq V$，所有满足$I \subseteq K \subseteq J$的$K\in\binom{V}{k}$构成集合的特征函数被称为“基本函数”。本文研究由两个“对立”基本函数相加得到的构造，在几乎所有情形下，这些“配对”函数都是布尔函数。\n我们的主要结果是确定了所有配对函数的精确次数——该次数是基于$n$元多项式表示定义的。证明过程是初等的，未涉及任何谱方法。首先，我们通过识别并结合所涉次数之间的多种关系，解决了$n=2k$的中间层情形。随后，我们通过导出函数、约化函数和对偶函数，将一般情形归约为中间层情形。\n值得注意的是，在某些情形下，该次数严格小于函数和的初等上界所保证的次数。这使得配对函数成为小支撑集固定次数布尔函数的优良候选。研究结果表明，当$n = 2k$且次数$t$为偶数且$t \notin \{0,k\}$时，配对函数提供了目前已知的最小非零布尔函数，超过了在所有其他情形下为已知最小构造的$t$-pencils。
英文摘要：
Let $V$ be a finite set of size $n$. We consider real functions on the "slice" $\binom{V}{k}$, which are also known as functions in the Johnson scheme. For $I \subseteq J \subseteq V$, the characteristic function of the set of all $K\in\binom{V}{k}$ with $I \subseteq K \subseteq J$ is called "basic". In this article, we investigate a construction arising as the sum of two "opposite" basic functions. In essentially all cases, these "paired" functions are Boolean. Our main result is the determination of the exact degree -- regarding a representation by an $n$-variable polynomial -- of all paired functions. The proof is elementary and does not involve any spectral methods. First, we settle the middle layer case $n=2k$ by identifying and combining various relations among the degrees involved. Then the general case is reduced to the middle layer situation by means of derived, reduced, and dual functions. Remarkably, in certain situations, the degree is strictly smaller than what is guaranteed by the elementary upper bound for the sum of functions. This makes paired functions good candidates for fixed-degree Boolean functions of small support size. As it turns out, for $n = 2k$ and even degree $t \notin \{0,k\}$, paired functions provide the smallest known non-zero Boolean functions, surpassing the $t$-pencils, which is the smallest known construction in all other cases.
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