关于多项式阈值函数的权和密度界
On the weight and density bounds of polynomial threshold functions
摘要：
本文证明了所有n元布尔函数都可以表示为至多具有$0.75 \times 2^n$非零整数系数的多项式阈值函数（PTF），并给出了这些系数的绝对值的上界.据我们所知，这提供了最好的已知的约束的PTF密度（单项式的数量）和重量（总和的系数幅度）的一般布尔函数。Bent函数的特殊情况也进行了分析，并表明，任何n-变量Bent函数可以表示为整数系数小于$2^n$，同时也遵守上述密度界。最后，稀疏布尔函数，这是几乎恒定的，除了变量分配的$m << 2^n$数，被证明有小的重量PTFs的密度在最多$m+2^{n-1}$。
英文摘要：
In this report, we show that all n-variable Boolean function can be represented as polynomial threshold functions (PTF) with at most $0.75 \times 2^n$ non-zero integer coefficients and give an upper bound on the absolute value of these coefficients. To our knowledge this provides the best known bound on both the PTF density (number of monomials) and weight (sum of the coefficient magnitudes) of general Boolean functions. The special case of Bent functions is also analyzed and shown that any n-variable Bent function can be represented with integer coefficients less than $2^n$ while also obeying the aforementioned density bound. Finally, sparse Boolean functions, which are almost constant except for $m << 2^n$ number of variable assignments, are shown to have small weight PTFs with density at most $m+2^{n-1}$.
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