# Angle-restricted sets and zero-free regions for the permanent

## TL;DR

This paper develops a systematic geometric approach to identify zero-free regions for the permanent of matrices, refining Barvinok's method by introducing angle-restricted sets, leading to improved bounds and explicit solutions.

## Contribution

It introduces angle-restricted sets and a geometric framework to construct zero-free regions for the permanent, improving upon Barvinok's previous results.

## Key findings

- Derived explicit zero-free regions for the permanent.
- Improved bounds over previous results.
- Reduced the problem to low-dimensional geometry.

## Abstract

The goal of this note is to give a systematic method of constructing zero-free regions for the permanent in the sense of A. Barvinok, i.e. regions in the complex plane such that the permanent of a square matrix of any size with entries from this region is nonzero. We do so by refining the approach of Barvinok, which is based on his clever observation that a certain restriction on a set S involving angles implies zero-freeness; we call sets satisfying this requirement angle-restricted. This allows us to reduce the question to a low-dimensional geometry problem (notably, independent of the size of the matrix!), which can then be solved more or less explicitly. We give a number of examples, improving some results of Barvinok.