Drew–Johnson–Olesky–van den Driessche superpattern conjecture for sign patterns

A sign pattern is a matrix whose entries are 00, ++, or -. It is spectrally arbitrary if every monic real polynomial of degree nn is the characteristic polynomial of an n×nn\times n real matrix obtained by replacing its nonzero entries with real numbers of the corresponding signs. A sign pattern is a superpattern of another if it is obtained by replacing some of the latter's nonzero entries by arbitrary nonzero entries while retaining its existing nonzero entries; a spectrally arbitrary sign pattern is minimal if none of its proper subpatterns is spectrally arbitrary. Drew–Johnson–Olesky–van den Driessche's superpattern conjecture. If SS is a minimal spectrally arbitrary sign pattern, then any superpattern of SS is spectrally arbitrary. This conjecture asks whether spectral arbitrariness is preserved when additional nonzero entries are added to a minimal spectrally arbitrary sign pattern. The source states that it remained open before this paper, whose abstract claims to provide a counterexample, so the conjecture is refuted.

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Primary source

Yaroslav Shitov, “Counterexamples on spectra of sign patterns”, arXiv:1612.05818 (2016).

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