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

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.

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