  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.
