# Diophantine equations with binomial coefficients and perturbations of symmetric Boolean functions

## TL;DR

This paper investigates the properties of perturbations of symmetric Boolean functions, linking exponential sums to Diophantine equations, and shows that balanced perturbations of fixed degree are rare as variables increase, providing specific examples and identities.

## Contribution

It extends the concepts of balanced Boolean functions to perturbations, proves non-existence of balanced fixed-degree perturbations for large variables, and uncovers unexpected identities.

## Key findings

- Balanced perturbations of fixed degree do not exist for large variable counts.
- Some sporadic balanced perturbations are identified.
- An unexpected identity between different symmetric Boolean function perturbations is presented.

## Abstract

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 $\Gamma$ 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.