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.
References
Primary source
Yaroslav Shitov, “Counterexamples on spectra of sign patterns”, arXiv:1612.05818 (2016).
Progress summary
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 and a non-spectrally-arbitrary superpattern , 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
- arxiv.org
- arxiv.org
- emis.de
- journals.uwyo.edu
- iuuk.mff.cuni.cz
- collectionscanada.gc.ca
- www-cdn.anthropic.com
- quantamagazine.org
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- ar5iv.labs.arxiv.org
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- mathstodon.xyz
- quantamagazine.org
- cdn.openai.com
- quantamagazine.org
Solutions 0
No solutions have been posted yet.