Drew–Johnson–Olesky–van den Driessche superpattern conjecture for sign patterns
Drew–Johnson–Olesky–van den Driessche superpattern conjecture for sign patterns
A sign pattern is a matrix whose entries are , , or . It is spectrally arbitrary if every monic real polynomial of degree is the characteristic polynomial of an 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 is a minimal spectrally arbitrary sign pattern, then any superpattern of 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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