二项式系数的丢番图方程与对称布尔函数的扰动
Diophantine equations with binomial coefficients and perturbations of symmetric Boolean functions
摘要：
本文研究了对称布尔函数的扰动问题。特别是，它建立了这些扰动的指数和之间的联系和丢番图方程的形式$$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$其中$x_j$属于一些固定的有界子集$Γ$的$\mathbb{Z}$。将平凡平衡对称布尔函数和偶发平衡布尔函数的概念推广到这类扰动。Canteaut和Videau对固定次数的对称布尔函数所作的一个观察是推广的。具体地说，证明了除平凡情形外，当变元数增加时，不存在固定次数的平衡扰动。一些零星的平衡扰动。最后，一个美丽的，但意想不到的身份之间的扰动两个非常不同的对称布尔函数也包括在这项工作中。
英文摘要：
This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form $$ \sum_{l=0}^n \binom{n}{l} x_l=0,$$ where $x_j$ belongs to some fixed bounded subset $Γ$ of $\mathbb{Z}$. The concepts of trivially balanced symmetric Boolean function and sporadic balanced Boolean function are extended to this type of perturbations. An observation made by Canteaut and Videau for symmetric Boolean functions of fixed degree is extended. To be specific, it is proved that, excluding the trivial cases, balanced perturbations of fixed degree do not exist when the number of variables grows. Some sporadic balanced perturbations are presented. Finally, a beautiful but unexpected identity between perturbations of two very different symmetric Boolean functions is also included in this work.
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