# A formula for systems of Boolean polynomial equations and applications to computational complexity

## TL;DR

This paper introduces a new formula based on binary trees and De Morgan's duality for systems of Boolean polynomial equations, enabling complexity analysis and fixed parameter tractability results for problems like SAT and graph coloring.

## Contribution

It presents a novel formula for Boolean polynomial systems that facilitates complexity analysis and demonstrates fixed parameter tractability for NP-complete problems.

## Key findings

- The formula has a binary tree structure and adheres to De Morgan's duality.
- It proves a complexity result parameterized by bandwidth.
- Shows NP-complete problems are fixed parameter tractable by bandwidth.

## Abstract

It is known a method for converting a system of Boolean polynomial equations to a single Boolean polynomial equation with less variables. In this paper, we show a formula for systems of Boolean polynomial equations which is based on the method. The formula has a structure of binary tree, and conforms to De Morgan's duality. Using the formula, we prove a computational complexity result with a parameter for solving systems. The parameter is the bandwidth in matrix and graph theories: to be precise, the definition follows convention in matrix and the value depends on the order of variables. We also apply the result to the NP-complete problems, SAT and graph list-coloring, to show that these problems are fixed parameter tractable by bandwidth.