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

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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.

References

Primary source

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

Progress summary

Refreshed
Claimed solved

A 2016 paper claims a counterexample disproving the conjecture, but the supplied record contains no independent verification.

The conjecture, proposed by Drew, Johnson, Olesky, and van den Driessche in 2000, asserts that every superpattern of a minimal spectrally arbitrary sign pattern is spectrally arbitrary.

December 2016 counterexample claim

The abstract of Counterexamples on spectra of sign patterns claims an example with a spectrally arbitrary pattern SS and a non-spectrally-arbitrary superpattern S′S', which would refute the conjecture. The supplied sources provide no independent verification, explicit-pattern analysis, or later error report.

Current status (as of September 2026): The conjecture is claimed to be refuted by a December 2016 paper, but that counterexample remains unverified in the retrieved record.

Sources

Solutions 0

No solutions have been posted yet.